![viii Preface
We should point out that the bibliography is far from being complete. We included
only the papers that we had studied during the preparation of the book or before. So
we have to apologize if readers are interested in historical developments of presented
results.
The theory of extension of holomorphic functions contains a lot of questions which
are still waiting for being solved. We have put many of them into the text (marking
them by [T]). The reader is encouraged to work on some of them.
During the process of proofreading we got the impression that this became a never-
ending story and there had to be a time for us to stop. We would be pleased if the
reader would inform us of any errors he/she may have detected while studying the
text.
It is our deep pleasure to thank our teachers, Professors H. Grauert and J. Siciak,
who taught us the beauty of complex analysis. We would also like to thank Dr. habil.
W. Zwonek who helped us in corrections of the text.
We thank the following institutions: Committee of Scientific Research (KBN),
Warsaw (PB 2 P03A 060 08, PB 2 P03A 017 14), Volkswagen Stiftung (Az. 1/71 062,
RiP-program at Oberwolfach), and Niedersächsisches Ministerium für Wissenschaft
und Kunst (Az. 15.3-50 113(55) PL). Without their financial support this work would
have never been possible. We would also like to thank our universities for support
during the preparation of the book.
Finally we thank Walter de Gruyter Publishers, especially Dr. M. Karbe, for having
encouraged us to write this book.
Krakow — Oldenburg, May 2000 Marek Jarnicki
Peter Pflug](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-12-320.jpg)
![Chapter 1
Riemann domains
Roughly speaking, a Riemann domain over Cn
is a 'domain spread over Cn
i.e.
a connected complex manifold X having a global projection ρ: X —> C" such that
ρ is locally homeomorphic. The class of Riemann domains over Cn
extends the
class of subdomains of Cn
. Riemann domains appear in a very natural way while
discussing problems related to holomorphic continuation. Consider the following
classical example. Let G := C and let Log denote the principal branch of the
logarithm. Then the function Log extends holomorphically to a domain X which is
no longer a plane domain but a so-called Riemann surface of the function Log. The
same phenomenon appears if we consider on G a branch of the k-th root. This means
that the maximal domain of existence of a holomorphic function defined in a plane
domain may be non-univalent — it may be a Riemann domain over C.
More generally, given a domain G c C", we can consider the maximal domain
of existence of a family 4 c 0(G), where 0(G) denotes the space of all functions
holomorphic on G. The most interesting case is the case when S = 0(G). From the
point of view of the theory of holomorphic functions the following three fundamental
questions are the most important:
(1) Does the maximal domain of existence always exist in the category of Riemann
domains over Cn
?
(2) What is a characterization of those domains which cannot be holomorphically
extended to any larger Riemann domain ?
(3) When is the maximal domain of existence univalent, i.e. can it be realized as a
domain in C" ?
It is well known that an arbitrary domain G C C is the maximal domain of
existence of the whole space 0(G), i.e. each boundary point of G is a singular point
for a function holomorphic in G. This is no longer true in C" with η >2. For example,
consider the following domain (cf. [Sha 1976]):
Let Ε denote the unit disc. Put Ρ := Ε χ (2Ε) c C2
and let
G := Ρ (Qi U 02 U S),
^ For the notation used in this book see 'List of symbols'.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-15-320.jpg)
![1.1 Riemann domains over C" 3
1.1 Riemann domains over C
Definition 1.1.1. A pair (Χ, ρ) is called a Riemann region (spread) over C" if:
• X is a topological Hausdorff space,
• ρ: X —> Cn
is locally homeomorphic.
If X is connected, then we say that (X, p) is a Riemann domain over Cn
. If (X, p) is a
Riemann region over C", then for any connected component C of X the pair (C, pc)
is a Riemann domain over Cn
.
If we replace in the above definitions the space C" by a connected «-dimensional
complex manifold Μ 3
then we get the notions of a Riemann region over Μ and
Riemann domain over M, respectively.
The mapping ρ is called the projection. For ζ € p(X) the set is called the
stalk over z. A subset A C X is said to be univalent (or schlicht) ii p A - A —• p(A)
is homeomorphic.
Observe that if A C X is open or compact, then A is univalent iff PA is injective.
Notice that, in general, a set A C X such that PA is injective need not be univalent
(Exercise).
In the sequel we will sometimes identify X with (X, p) if it is clear from the
context what the projection ρ is.
Remark 1.1.2. (a) If G c Cn
is a domain, then (G, idc) is a Riemann domain over
CM
. This will be the standard identification of domains in C" with Riemann domains.
Of course, any locally homeomorphic mapping ρ: G — • Cn
makes from G a
Riemann domain over C". For example, (C, exp) is a Riemann domain over C.
(b) If (X, p) is aRiemann domain over Cn
, then ρ is an open mapping. In particular,
the set p(X) is a domain in Cn
. For any ζ e p{X) the stalk p~x
{z) is discrete.
(c) If (X, p) is a Riemann domain over C , then the family (U, pu)u. where U
2
> That is, each point a € X has an open neighborhood U such that p(U) is open in C" and
pu U —• p(U) is homeomorphic.
3
' Recall (cf. [Nar 1968]) that a topological Hausdorff space Μ is called an n-dimensional
Ck
-manifold (resp. complex manifold) if there exists a family (£/,, φ,· ),·<=/ (called an atlas) such that:
• (^i)ie/ is an open covering of M,
• φι is a homeomorphic mapping of £/; onto an open subset φι (ί/, ) of R" (resp. C"),
• for any i, j e I, if t/, Π Uj Φ 0, then the mapping ψ] ο <pfl
is of class Gk
(resp. holomorphic)
on <pi{Ui Π Uj)here k e Z+ U {oo} U {ω}, where ω real analytic.
It is clear that any complex «-dimensional manifold is a 2n-dimensional real analytic manifold.
Moreover, any Cl
-manifold is an C*-manifold for 0 < k < £ < ω (with oo < ω).
Suppose that Ν is another m-dimensional -manifold (resp. complex manifold) with an atlas
(Vj, ipj)jej. We say that a continuous mapping F: X —> Y is of class Gk
(resp. holomorphic) if
• for any i e I, j e J, the mapping ψ} ο F ο φ7ι
is of class Gk
(resp. holomorphic) on
<p, (t/,· Π F"UVj)) in the classical sense; we write: F e Ck
(M, N) (resp. F e 0(M, N)). Put
Ck
(M) := Ck
(M, C), Θ{Μ) := Θ(Μ, C).
Notice that any connected manifold is arcwise connected.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-17-320.jpg)
![1.1 Riemann domains over C" 5
(b) Let A i, A2 C X be univalent sets such that
• ΑχΓΑ2φ0,
• the set p(A) Π p(A2) is connected-
Then the set Αι U A2 is univalent.
Notice that the conclusion of (b) does not hold if A Π A2 = 0 or p ( A ι ) Π p(A2 )
is disconnected.
Proof, (a) follows directly from the identity principle.
(b) Fix an x0 e At Π A2. Let Τ := p(A) Π p(A2), Sj := {ρΑ])~Χ
τ, j =
1, 2. Observe that si and s2 are sections over Τ such that s(p(xo)) — s2(p(xo)).
Consequently, by (a), s = s2, which finishes the proof. •
Let (X, p) be a Riemann domain over C". Since ρ is locally homeomorphic, we
can introduce on X the notion of a ball. More precisely, let
q : C" — • M+
be an arbitrary C-norm. Put
Bq(zo, r) := {z e Cn
: q(z - z0) < r}, z0 e Cn
, r e (0, +oo], 5)
For a e X and r e (0, +oo] let
Bq(a, r) = Bx^(a, r)
denote an open univalent neighborhood of a (if it exists) such that p(Bq(a, r)) =
Bq(p(a),r). Observe that:
• Bq(a, r) exists for small r > 0;
• if Bq(a, r) exists, then it is uniquely determined;
• if Bq(a, r) exists, then for any a' € Bq(a, r) andO < r' < r — q(p(a') — p(a)),
the ball Bq (a', r') exists and
Bq(a',r') = {p^iar))-Bq{p{a'),r'))·
• if Bq(a, r') exists for any 0 < r' < r, then Bq(a, r) exists and
Bq(a,r)= (J Bq{a,r').
0 <r'<r
In particular, if we define the distance to the boundary (w.r.t. the norm q)
dx,q '
• X — • (0, +oo],
dx,q{a) := sup{r G (0, +oo]: Bq(a, r) exists}, a e X,
5) Notice that Bq (z0, +oo) := C".](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-19-320.jpg)
![1.1 Riemann domains over C 7
In particular, the function άχ,q is continuous.
Λ. A. A
Indeed, by Corollary 1.1.6(b), for any χ g Z?q (α) the set U := ßq (a) U Bq(x) is
a univalent subdomain of X. Consequently, we have:
dx,q(x) - dx,q(a) = ^([/.plyJ.qU) - ^(t7,p|c/),q(«)l
= M(P(t/),id),q(pU)) -d(p(t/),id),q(/?(a))l < q(/>(*) - p(a)).
(d) If (Xj, pj) is a Riemann domain over Cn
>, qj : — • R+ is a C-norm,
j = 1,2, andq(zi,z2 ) := max{qi(zi), q2(z2)h then
d(x1xx2,pixp2),q(xi, X2) = min{d(xliPl),qi (*i), </(χ2 (*2)},
{x,xi) G Xi X x2.
(e) Let Κ c X be compact. Then:
• dx,q(K)> 0.
• the set /sT(r,<,)
is compact for any 0 < r < dx<q(K).
Indeed, let ι C *υ g Bq(av,r), av g Κ, ν > 16)
. Since /Γ is
compact, we may assume that av —> ao e Κ and p(xv) —> zo <ξ C". Moreover, we
may assume thatav G Bq(ao, ε), ν > 1, where ε := dxq{K) — r. Then Bq(av, r) C
Bq(ao) and, consequently, = p^]q(p(xv)) — • P^]q(zo) =• *o- Moreover,
q(p(*o) - p(ao)) < r. Hence xo G Bq(a0, r) C
• dx,q(K^) > dx,q(K) - r (cf. Corollary 1.1.10).
(f) If Κ is compact and univalent, then K(r
'q
* is univalent for small r > 0.
Indeed, suppose thatxv g Bq(av, 1/v), yv G Bq(bv, 1/v), av, bv g K,xv ^ yv,
p(xv) = p(yv), ν » 1. We may assume that av —> ao e K, bv — • bo G K. Then
p(ao) = ρ (bo) and therefore ao = bo. Thus xv, G Bq(ao), ν 1; contradiction.
Proposition 1.1.9. IfX
oo = 0, thenfor any a G X there exists a vector w G Cn
with
q(w) = dx,q(a) such that
lim dx,q((pa,q)~l
(p(a) + tw)) = 0.
[U, 1 • 1
In particular, dx<q(Bq(a)) = 0.
Proof. Fix an a G X. To simplify notation put
Λ A
d:=dx<q, Uo:=Bq(a),
i/o := Bq(p(a),d(a)) = p(Ü0), s := (ρα,μΓι
= (ρόοΓ1
.
6)
One can prove (cf. Remark 1.1.14) that X is metrizable, and therefore, the compactness of
JS-(r, q)may be checked via sequences.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-21-320.jpg)
![1.1 Riemann domains over C" 9
C
o
r
o
l
l
a
r
y 1
.
1
.
1
0
. Let Κ c X be compact. Then
dx,q{K^) = dx,q(K) - r, 0 < r < dx,q(K)
(cf. Remark 1.1.8(e)).
Proof. We may assume that Xoo — 0· Fix Κ and r. We already know that
dx,q(K(r
'q
^) > dx<q(K) — r. Since the function άχΛ is continuous, there ex-
ists an a e Κ such that dx,q(a) — άχ^(Κ) =: R. By Proposition 1.1.9 there
exists a w e Cn
with q(u;) = R such that lim[o,i)3f^i dx,q(x(t)) = 0, where
x(t) := (pa,q)~l
(p(a) + tw). Let θ e (0, 1) be such that r = 6R. Put b := x(ß).
Then b e K(r
'q)
and x(t) e Bx<q(b, R - r) for θ < t < 1. Consequently,
dx,q(b) = R-rmd hence dx<q(K^^) = dx,q{K) - r . •
Remark 1
.
1
.
1
1
. Let γ: [
0
,
1
] —> p(X) be a curve. Fix an a e ρ~1
(γ(0)) and let
Jo denote the set of all f e [0, 1] such that there exists a curve yt: [0, /] —> X with
yf(0) = a and ρ ο yt = y |[o,r]- Then we have:
• 0 e /o.
• If t e /o, then [0, t] c Iq and therefore Iq is an interval.
• yf/ = YT»[0J>] for 0 <t'< t", t" e IQ (by the identity principle for liftings) and,
therefore, we have a lifting γ: IQ —> X of γ |/0 with p(0) = a.
• IQ is open in [0, 1]. Indeed, if U is an open univalent neighborhood of Y(t) (for
some ί e /ο Π [0, 1)) and 0 < δ < 1 - ί is such that γ ([ί, t + δ]) C p(U), then we
can define:
Μ ) , t g [0, t]
(.pu)-l
(y(t)), te[t,t + S].
Consequently, either IQ = [0, 1] (and then γ is a lifting of y ) or IQ = [0, t*) for some
0 < t* < 1.
• If IQ = [0, /*), then dx(y(IO)) = 0 (in particular, y(IO) is not relatively com-
pact). Indeed, suppose that άχ(γ(Ιο)) = r > 0. Take a point 0 < t < t* such that
y(t) € P(y(t), r) for t € [t, i*] and define
γ tit), t e [0, t]
Yt+s(0 := { ,„,.Λ _ι(
_ I Y i ( t
l Pöi
Then yt* is well defined and it is a lifting of / |[o,f*]; contradiction.
Analogously as we defined the distance to the boundary (w.r.t. a norm), we can
introduce the distance to the boundary in a given direction.
For ζ ε Cn
, ξ e Cn
, and r > 0 let
A^z,r) :=ζ + (ΓΕ)ξ
(note that Δ0(ζ, r) := {z}).](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-23-320.jpg)
![10 1 Riemann domains
For a point a e X let Αξ(α, r) denote a univalent set containing a (if it exists)
A
such that ρ(Αξ(α, r)) = Αξ(ρ(α), r).
A A
Observe that the disc Αξ (a, r) exists for small r > 0. Moreover, if Αξ (a, r) exists,
then it is uniquely determined (note that Δο(α, r) = {a}).
Define the distance to the boundary in direction ξ, δχ^ : Χ —• (0, +oo],
A
δχ,ξ(α) := sup{r > 0: Αξ(α, r) exists}, a € X.
A A A A
Remark 1.1.12. (a) If Αξ(α, r) and Αη(α, s) exist, then the set Αξ(α, r) U Αη(α, s)
is univalent (use Corollary 1.1.6(b)).
(b) If Αξ(α, r) exists (ξ φ 0), then for any a! € Αξ(α, r) and 0 < r' < r —
|| p(a') - p(a)||/||£||, the disc Ας (a', r') exists and
Α ξ ( α ' , / ) = ( ρ λ ξ Μ ) - ί
( Α ξ ( ρ ( α ' ) , / ) ) .
If Αξ(α, r') exists for any 0 < r' < r, then Αξ(α, r) exists and
Δ ξ(α,Γ)= ( J Α ξ ( α , / ) .
0 <r'<r
A A
(c) Assume that Δξ0(αο, R) exists. Fix 0 < ro < R. Put Κ := Αξ0(αο, ro) and
let ε > 0 be so small that the set Λ := is univalent (cf. Remark 1.1.8(f)). Let
U c X x C" χ R>o be an open neighborhood of the point (ao, ξο, ro) such that
A$(p(a),r) C p(A) for any (a,%,r) 6 U. Then for any (a,£,r) e U the disc
Αξ(α, r) exists and
Δί (α,Γ) = (ρ|Α )-1
(Δξ (ρ(α),Γ)).
In particular, the function
X xCn
3 ( χ , ξ ) ^ δχ,ξ(χ) e (0, + o o ]
is lower semicontinuous.
(d) Let q: Cn
—> R + be a C-norm. Then Bq(a, r) exists iff Αξ(α, r) exists for
any ξ with q(£) = 1. Moreover,
Bq(a,r)= (J Αξ(α,Γ).
q«)=l
Hence
d x , q ( x ) = Μ{δχ,ξ(χ): ξ e C", q(!;) = 1}, χ 6 X.
Notice that Proposition 1.1.9 says that for any a e X there exists a ξ € C" with
q(£) = 1 such that dxiq(a) = δΧιξ(α).](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-24-320.jpg)
![1.1 Riemann domains over Cn
11
Lemma 1.1.13. Let ξ e Cn
, ||£|| = 1, and let
q^(z):=max{±||z-<z,£)£||, (ζ,ξ)1 ze C (e > 0),
where { , ) denotes the standard Hermitian scalar product in C" Then
dx ,qfi£ / * δχ,ξ when ε 0.
Proof. First observe that if L: Cn
—> Cw
is a C-linear isomorphism, then
d(X,p),q = d(X L o p ) i t l o L -l, δ(Χ,ρ),ξ = hx,Lop)MH)·
If, moreover, L is unitary and L(£) = en = (0,..., 0, 1), then
q^e ο L-1
= qen,e·
Thus we may assume that ξ = en.
Obviously, dXi({en e < Sx,£n and dx, „ ^ , > dXt q ^£ „ when 0 < ε' < ε".
Now fix an a e X, take 0 < r < δχ,βη(α), and let Κ := Αβη(α, r). Since Κ is
compact and univalent, there exists 0 < η < dx(K) such that the set A := is also
univalent. It is clear that there exists an ε > 0 such that U := Bqen ε(ρ(α), r) c ρ (A).
Hence (pA)~l
(U) = Bx^en e(a, r) and, consequently, dx^en C(a) > r. •
For any curve γ: [0, 1] — • C" let L(y) denote the length of γ (w.r.t. the Euclidean
norm), i.e.
Ν
L(y) := sup { Σ IIY(tj) - Y(tj-1)||: Ν e N, 0 = t0 < • • • < tN = l}.
j=ι
Now we introduce a structure of a metric space on a given Riemann domain (X, p).
First, since X is arcwise connected, we conclude that for any x', x" e X there exists
a curve γ: [0, 1] —> X with y(0) = x', y(l) = x" such that L(p ο γ) < +oo.
Indeed, if γ: [0,1] —> X is an arbitrary curve with p(0) = x', p(l) = x", then we
A
find points x,..., xn-1 £ γ([0, 1]) such that xj- € M(xj), j = 1,..., N, where
XQ := x', XN '•— X
"• The curve defined as the union of the 'segments'
is well defined and L(p ο γ) = Σ]=ι p(xj~i) — p(xj) < +oo.
Now, we define the inner (arcwise) Euclidean distance on (X, p)
σχ: Χ χ X —• R+
by the formula
σχ{χ', χ") := inf{L(p ο γ):
γ: [0, 1] — • Χ is a curve with p(0) = x', y (1) = χ"), χ', χ" € X.
7)
(z,w) :=Ej=iZjWj.
^ Observe that ζ — (ζ, ξ)ξ is the orthogonal projection of the vector ζ on the complex hyperplane
(Q)-1
-. Note that is a C-norm.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-25-320.jpg)
![12 1 Riemann domains
Remark 1.1.14. (a) σ^ρ)(χ', χ") > σ^χ)^){ρ{χ'), Ρθ"))> x" e X.
(b) If A C X is a univalent set such that ρ (Λ) is convex, then
σχ{χ',χ") = ||pix') - p(x"), χ', χ" € Α.
Λ
(c) Βχ(α, r) = {λ € Χ : σχ(α, χ) < r}, a e Χ, 0 < r < ρχ(α).
Indeed, the inclusion ' c ' follows from (b). Suppose that σχ(α,χ) < r. Then
there exists a curve γ : [0, 1] —• X with γ(0) = α, γ(1) = χ, and L(p ο γ) < r. In
particular, (ρ ο y)([0,1]) C M(p(a), r). Hence, by the identity principle for liftings,
Υ '• [0, 1] —• Βχ(α, r).
(d) σχ is a continuous distance. Observe that the topology generated by σχ coin-
cides with the initial topology of X.
(e) For any metric space (7, d) and mapping f : X —> Υ the following conditions
are equivalent:
(i) d{f(x'),f(x"))<ax{x',x"), x',x"eX
(ii) d(f(x'),f(x))<p(x')-p(x), χεΧ,χ'eMx(x).
In particular, by Remark 1.1.8(c), if Χ<χ, = 0, then
Ipx(x') - pxix")I < σχ(χ', χ"), χ', χ" € Χ.
The implication (i) (ii) follows from (b). Suppose that / satisfies (ii). Fix
x', x" G X and a curve γ: [0, 1] —• X with γ(0) = λ', y(l) = χ". It is clear
that there exist Ν e Ν and 0 = fo < · · · < = 1 such that γ(ί/) e Βχ(/(ί,_ι)),
j = 1,..., N. Hence
Ν
d(f(x'), fix")) < ^d(f(Y(tj-1)), f(Y(tj)))
7=1
Ν
< Ε WpiyVj-M ~ p(y(tj» < l
(P ° y).
j=ι
which implies (i).
(f) Let ao ε X, 0 < ε < ρχ (üq) and let Ue be the connected component of the
set {x e Χ: ρχ (χ) > ε} with üq g Ue. Let συε be the inner Euclidean distance on
(t/e, pue)· Then for any t > 0 the set
Ue,t := {x eUg·. aUe(ao,x) < t}
is relatively compact in X.
Indeed, let I := {t > 0: U£J m X}. Observe that, by (c), (0, ε] c /. To prove
that I = R>o it suffices to check that if f e I, t > ε, then t + ε/2 e I. Thus, it
suffices to show that
Ue,t+e/2C U Βχφ,ε).
beU£,t](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-26-320.jpg)
![1.1 Riemann domains over Cn
13
Take a point a e Uej+e/2 Uej and let γ: [0,1] —• Us be a curve with p(0) = ao,
γ(1) — a, and L(p ο γ) < t + ε/2. Take a r € (0, 1) such that the point b := γ (τ)
lies in Βχ(α, ε) ε/2). Then
ί + ε/2> L(p ογ) = L(p ο γ|[0,τ]) + L{p ο p|[r,i])
> L(p ο y |[o,r]) + IIΡΦ) - Ρ(α)II > L(p ° Pl[0,r]) + ε/2.
Consequently, L(p ο γ|[ο,τ]) < t. Hence b e U£it and a e Βχ(£, ε).
Remark 1.1.15. (a) It is known (cf. [Die 1944], [Bou 1965]) that for any connected
manifold Μ the following conditions are equivalent:
(i) Μ is countable at infinity i.e. Μ = (J/^i where Kj is compact, j —
1 2 9
>·
1, i.,... ,
(ii) The topology of Μ has a countable base.
(b) If Μ is a connected Riemann surface (i.e. a connected one-dimensional complex
manifold), then Μ is countable at infinity; cf. [Rad 1925], see also § 1.11. Theorem
1.1.17 will show that the same property is shared by all Riemann domains over C".
Exercise 1.1.16. Complete the following construction showing that there exists a
simply connected two-dimensional complex manifold Μ which is not countable at
infinity (cf. [Cal-Ros 1954]).
Let A c C, A uncountable, Μ := A χ C2
,
top Μ := {U CM: VfeA: {(x,y) e C2
: (t,x,y) € U} etopC2
}.
Fori, t g A, s φ t, put
<DJjf: C χ C* —> C χ C*, Φ5>ί(χ, y) := +*,?);
Φίι? is biholomorphic and Φ"/ = ΦΜ . For (s, xs, yj), (t, xt, yt) G Μ define
(s,xs, ys) ~ (t,xt, yt)
ys) = (t,xt, yt)) or (s φί, ys = yt, s+xsys = t + xty?)
((j, ys) = {t, xt, yt)j or ( j φ t, ys, yt e C*, (xs, ys) = Φs,t(xt,
~ is an equivalence relation. Put Μ :— Μ/ ~ and let top Μ be the standard quotient
topology, i.e. the strongest topology on Μ such that the canonical projection
Μ 3 (t,x,y) [(*,*,?)] e Μ
^ Observe that, since Μ is locally compact, the following conditions are equivalent:
• Μ is countable at infinity;
• Μ = U/^i Uj, where Uj is relatively compact and open, j = 1,2,...;
• Μ — U/ÜLi where Kj is compact and Kj C int Kj+, j — 1,2,](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-27-320.jpg)
![14 1 Riemann domains
is continuous;
U e top Μ <(=?• VfeA: {(*,?) € C2
: [(?,*,>>)] € U) € topC2
.
For t e A define
Ut := Tim x C2
), φ,: C2
—> t/„ <p,(x, y) := [(ί, λ, y)].
Then:
• (pt is continuous and bijective.
• U e top Μ Viey4: (£/ Π I/,) 6 top C2
.
• Ut € top M.
• is continuous.
. M = U ( e A [ / ( , t o p M 6 r 2 .
• (i/r, is an atlas of a complex structure on M.
• Μ is connected.
• The set S := π {A χ {(0,0)}) is discrete and #5 = #A. In particular, Μ cannot
be countable at infinity.
• Μ is simply connected.
Theorem 1.1.17 (Poincare-Volterra theorem). Let (X, p) be α Riemann domain over
Cn
. Then X is countable at infinity. In particular,
• for each ζ € p(X) the stalk p~l
(z) is at most countable;
• the topology ofX admits a countable base consisting of connected univalent and
relatively compact sets.
Proof. Fix a point oq G X and let Us and UEt, be as in Remark 1.1.14(f). Let ko € Ν
be such that 1/ko < ρχ(αο). Put
Vit := Ui/k,k, k > k0.
By Remark 1.1.14(f) V* ^ X, k > ko. Obviously V^ c V^+i. It remains to observe
thatx = u
r
=
*
0 V 0 )
.
Since X is countable at infinity, it is clear that each stalk ρ 1
(ζ) must be at most
countable. Now we will directly show that the topology of X admits a countable
base consisting of connected univalent and relatively compact sets.
Let X = U ~ i where Kj is compact, j e N. For any j and for any rational
number 0 < r < dx(Kj), consider the covering (P(x, r))x€Kj of Kj. Since Kj is
A
compact, we can select a subcovering (P(x, r))x€sjr, where c Kj is a finite set.
We claim that the family
05 := (P(x, r));eNi ο<r<dx{K}), reQ, xeSj,r
Take an arbitrary point a E X and a curve Γ: [0, 1] —> X with p(0) = OQ, y(l) = A,
i := L{p ο γ) < +oo. Let 0 < ε < ρχ(γ([0, 1])). Then a e ί/ε>ί with t > I. Consequently, a e V^
with k > max{/co, 1/ε, I}.
1
^ Independently of Remark 1.1.15(a).](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-28-320.jpg)
![1.1 Riemann domains over C" 15
is the base we are looking for. We only need to prove that 05 is a base of the topology
of X. Take an arbitrary open set U C X and a point a e U. Let 0 < R < άχ(α)
be such that P(a, R) C U. Let jo be such that a e Kj0 and let 0 < ro < dx(Kj0),
ro € Q, be such that ro < R/2. There exists an xo € Sy0>/-0 with a e P(;co, ro). It
A A
remains to observe that P(xo> r
o) C P(a, R) C U. •
The end of this section is devoted to continuity properties of families of liftings.
Theorem 1.1.18 (Monodromy theorem). Let (X, p) be α Riemann domain over Cn
.
Let A be locally compact, connected, and locally connected topological space and let
Β be a locally connected topological space. Fix an ao € A. Assume that
F Αχ Β —• p(X)
is a continuous mapping. Suppose thatfor each b e Β the mapping Ft, := F(·, b) has
A A
a lifting Fb: A —> X such that the mapping Β 3 b —> />(ao) e X is continuous.
Then the mapping
Α χ Β 3 (a, b) —Fb(a) e X
is continuous.
Proof. First observe that F is continuous at each point (ao, b), b e B.
Indeed, fix a b e Β and let W be a univalent neighborhood of F(ao, b ). There
exist connected neighborhoods U c A of ao and V c Β of b such that:
F({ao] χ V) c W (here we use the continuity of the mapping b —> F(ao, b)),
F(U xV)c p(W).
Define σ := (ρψ)~1
°Fuxv • Then for arbitrary b € V the mappings U 3 a —•
F(a,b) and U 3 a —> a(a,b) are liftings of the mapping U 3 a —> F(a, b) that
coincide at a = ao- Consequently, by the identity principle for liftings, they coincide
A A
everywhere. This means that Fuxv = o. In particular, F is continuous at (ao, ^l)·
Now, fix an arbitrary point (a, b) e Α χ Β. We will prove that there exist
neighborhoods U C A of at V c Β of b, W C X of F(a, b) such that W is
univalent, F(U χ V) c p(W) and Fuxv = (plw)- 1
° FuxV (which obviously
implies the continuity of F on U χ V).
For any χ e A let Wx C X be an arbitrary univalent neighborhood of Fb^x).
Choose a relatively compact neighborhood Üx c A of χ such that Ft>l (Üx) (s Wx.
Since F is continuous at (ao, b), we may assume that F(Üao χ V) C Wao for a
neighborhood V c Β of b. Since A is a connected and locally connected Hausdorff
space, there exist points x,..., χχ e A and domains U,..., U^ such that jci = ao,
ai 6 UN, Uj C ÜX], j = 1,..., N, and Uj nUj+i φ 0, j = 1,..., Ν - 1 (cf.
[Bou 1965]). Put Wj := WXj, j = ,...,N. Since F(Uj χ {^i}) = p{Fbx{Vj)) m
p(Wj), j = 1,..., N, there exists a connected neighborhood V C V of b such
that F(Uj χ V) c p(Wj), j = 1 ,.,.,Ν. Note that F{UX χ V) C W. Define
Oj = (pwj)~l
° Fujxv Uj χ V —> X. It suffices to show that for any
j = l,...,N:
σ
ί = FUjxV- (*)](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-29-320.jpg)
![16 1 Riemann domains
Observe that σ) = F on Uj χ {b}, j = I,..., N. Obviously (*) holds for j = 1.
Suppose that (*) is true for j = ... ,k for some 1 < k < Ν — 1. Fix a point
ε Uk Π Uk+i. Note that F(xo, b) = a^ixo, b), b e V. Fix a b e V. Observe
that the mappings Uk+1 3 λ: —> F(x, b), Uk+ι 3 x —> Ok+{x, b) are two liftings
of the mapping [/*+1 3 x — • F(x,b). Thus, by the identity principle for liftings,
it suffices to prove that a^ixo, b) = F(xo,b) = σ*+ι(χο- b). For, observe that the
mappings V 3 y —a^ixo, y) and V 3 y —> σ£+ι(χοι >0 are two liftings of
V 3 y —> F(xo, y) that coincide at y = b. Hence, once again by the identity
principle for liftings, we get the required equality. •
In the case where A = Β := [0, 1], ao := 0 we get the following
Theorem 1.1.19 (Classical monodromy theorem). Let (Χ, ρ) be α Riemann domain
over Cn, xo ^ X' and let F: [0, 1] χ [0, 1] —> p(X) be a continuous mapping such
that F{ 0, u) — p{x o), u € [0, 1]. Suppose thatfor each u € [0, I] there exists a lifting
Yu [0, 1] —> X of the curve F(·, u) such that y«(0) = xo. Define F(t, u) yu(t),
Λ
t,u 6 [0, 1]. Then F is continuous.
In particular, if there exists a point z € p(X) such that F(l,u) = z for all
λ -j
u e [0, 1], then the mapping [0, 1] 3 u —>· F(l,u) € ρ (zi) is continuous.
Consequently, since the stalk p~l(zi) is discrete, there exists a point X[ € p~^(z)
such that F{, u) = x for all u e [0, 1].
Α Riemann domain (X, p) is said to be a cover if for any point zo € p(X) there
exists an open neighborhood UQ such that each connected component U of the set
p'HUo) is univalent and p(U) = UQ.
Α Riemann domain (X, p) is said to be arbitrarily continuable if for any curve
γ: [0,1] —> p(X) and for any a e ρ~ι(γ(0)) there exists a lifting γ: [0,1] —> X
with y(0) = a.
Notice that in fact (X, p) is arbitrarily continuable iff (X, p) is a cover; cf. the
following exercise.
Exercise 1.1.20. Let (X, p) be a Riemann domain over C". Prove that the following
conditions are equivalent:
(i) (X, p) is a cover;
(ii) (X, p) is arbitrarily continuable;
(iii) For any metric space Τ such that
• Τ is arcwise connected,
• Γ is locally arcwise connected,
• Τ is homotopically simply connected,
for any continuous mapping γ: Τ —>· p(X), for any ίο € Τ, and for any
a e p~l(y(to)), there exists a lifting γ: [0, 1] —> X such that γ (to) = a
Notice that in (ii) we have simply Τ : = [0, 1].](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-30-320.jpg)
![1.1 Riemann domains over C" 17
(iv) For any C-norm q: Cn
—> R+ we have d(x p ) ^ = (d(P(X),id),q) ο ρ (cf.
Remark 1.1.8(a)).
Hint, (i) =>• (ii): Take an arbitrary curve γ: [0,1] —> p(X) and a point a e
ρ~ι
(γ(0)). Let /o be as in Remark 1.1.11. Since (X, p) is a cover, the set /o is
closed. Hence Iq = I.
(ii) =Φ· (iii): For any t e Τ let at: [0, 1] — • Τ be an arbitrary curve with
σ,(0) = ί0» σ,(1) = t- Put 8t := γ ο σ,: [0, 1] — • p(X) and let St: [0, 1] —> X
denote the lifting of 8t with <5,(0) = a. The classical monodromy theorem implies
that is independent of at . Define γ{ΐ) := <5,(1). Now it remains to prove that γ
is continuous.
(iii) (iv): Fix a C-norm q: C — • R+ and a e p_ 1
(zo)· Let
Τ := Bq(p(a), r) c p{X), γ := idr .
By (iii) there exists a section γ : Τ — • X of (Χ, ρ) over Τ such that γ (p(a)) = a.
Then y(T) = Z?q(a,r).
(iv) =>• (i): Fix a z 0 e p(X) and letl(z0 , 2R) c p(X). Put Uq := B(zo, r). Let
U be a connected component of p~l
(Uo). Then U = Μχ(α, r), where a e p~l
(zo)·
In particular, pu maps homeomorphically U onto Uq.
Notice that the assumptions on Τ may be weakened.
Proposition 1.1.21. Let (X, p) be a cover over C". Then:
(a) #p~x
{z) = #p~l
{zi) for any z, zi e p(X).
(b) If ρ(X) is simply connected, then ρ is injective (i.e. X is univalent).
Proof, (a) Fix Z,Z2 £ p(X), z φ Z2, and let γ: [0, 1] —> p(X) be a curve such
that y(0) = zi, y(l) = zi. It suffices to prove that #p~x
(z) < #p- 1
(^2)· For any
χ e p~l
(z) let f(x) = γχ(1), where γχ is the lifting of γ with yx(0) = x. It is clear
that / : p~l
(zi) —> p~l
(z2)· By the identity principle for liftings the mapping / is
injective.
(b) Suppose that αι,α2 € X are such that/?(a i) = p(a2) =: zoandleta: [0,1]—•
X be a curve such that σ(0) = αϊ, σ(1) = α2· Let
F : [0, 1] x [0, 1] —> p(X)
be a continuous mapping (a homotopy) such that
• F(0, .) = F(l,-) = zo,
• F(·, 0) = ροσ,
• F(; 1) = zo·
Since (X, p) is arbitrarily continuable, for each u e [0, 1] there exists a curve
F(·, u): [0,1] —> X such that F{0, u) — a. In particular, by the identity prin-
ciple for liftings, F(·, 0) = σ, F(·, 1) = ai and ρ ο F(-, ü) — F{-, u). By the
monodromy theorem, a2 = F( 1,0) = F(l, 1) = a. •](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-31-320.jpg)
![20 1 Riemann domains
Definition 1.2.4. Let !F c O(X) be a vector subspace. Assume that Τ is endowed
with a Frechet space topology τ. We say that F is a natural Frechet space if the
convergence in the sense of τ implies the convergence in the sense of τχ, i.e. the
identity embedding {Τ, τ) —(0(X), τχ) is continuous.
Obviously, (Θ(Χ), τχ) is a natural Frechet space.
The following example shows that many classical spaces of holomorphic functions
are natural Frechet spaces.
Example 1.2.5 (Natural Frechet spaces), (a) Jf°°(X) := the space of all bounded
holomorphic functions on X with the topology generated by the supremum norm
|| ||χ. Notice that (M°°{X), || ||χ) is even a Banach algebra.
(b) Lj(X) := Θ(Χ) Π LP (Χ) = the space of all p-integrable holomorphic func-
tions with the topology induced by the norm || lp(X) (1 < ρ < oo).
Here LP
(X) is taken w.r.t. the Lebesgue measure A — Αχ on X constructed as
follows: A set A C X is called measurable (A e £(X)) if p(A Π U) is Lebesgue
measurable in C" for any open univalent set U C X. It is clear that any Borel subset
of X is measurable. One can easily check that a set A c X is measurable iff any point
a e X has an open univalent neighborhood U such that p(A Π U) is measurable.
Let X = xUj, where each U/ is open and univalent (use the Volterra theorem).
Put Bi := Uu Bj := Uj {U U · • · U Uj-i), j =2,3,.... For A e £(X) we put
A*(A) := ΣΤ= ι A2n(p(A Π Bj)). One can prove (Exercise) that Αχ is a regular
Borel measure on £(X) which is independent of the choice of a sequence (i//)?^.
Moreover, Λχ(Α Π U) = Λ2η (ρ(Α Π U)) for any measurable set A c X and open
univalent set U c X. Observe that if A e £(X) and / : A —> [0, +oo] is a
measurable function, then fA fdAx = J^JLi ΙΡ(απβ ) f 0
(Puj)~l
dA2 n, where Uj
and Bj, j = 1, 2,..., are as above.
By the Cauchy integral formula, we get
UWK < — W f fdA, f G O(X), K m x , 0 < R < dx(K).
(πr1
)" JK(r)
Hence,
UWK < 2 B ll/ll£P(X), / e Lp
h(X), K m X , 0 < R < dx(K),
where l/p + 1/q = 1. In particular, L%(X) is closed in LP
(X), which shows that
{Lp
h{X), II ||lp(X)) is a natural Banach space. Obviously, L%>(X) = 3t°°(X).
Note that L2
h (X) with the scalar product
(/, «)—•/* fidA
Jx
is a Hilbert space.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-34-320.jpg)
![22 1 Riemann domains
A
The class [(i/, f)]a is called the germ of / at a. We write fa := [(£/, /)]a. Define
KU, f ) h + [(V, g)h ••= l(u nv,f + g)h,
[(£/, f ) h · [(V, g)h ·.= [(£/ η v, / · g)h.
One can easily check that the operations +, : Θα χ Θα —> Θα are well defined
and that (Θa , +, ·) is a commutative ring with the unit element (the ring of germs of
holomorphic functions at a).
Let f = fa e Θa. Then f(a) := f(a) and dv
f := [(U, 3v
/)]„ are well defined. In
particular, we can define
Taf(z) := T ^dv
f(a)(z-a)v
= Taf(z).
—' ν!
The mapping
Θα β f —> Τa f € the ring of all power series with center at a
that are convergent in a neighborhood of a
is an isomorphism. Put
Θ = (J Θα χ {a}
aeC
and let π: Θ —> C" be given by the formula 7r((f, a)) = a (in the sequel we will
denote elements of Θ either formally as pairs (f, a) (when we want to point out that
f € Θa), or simply as germs f). For f € Θα and for any (U, / ) € f define
Uf(U, f )•.={ ( f z , z ) : z e U ) c 0 .
It is clear that the system (ilf (U, / ) ) a € c n
, feöa , (t/,/)ef is a neighborhood basis 1 6
We endow Θ with the topology generated by this basis. We will show that this is a
Hausdorff topology. Take (f, a), (g, b) e Θ, (f, a) φ (g, b). If α φ b, thenllf (U, f ) Π
Ü0(V, g) = 0 provided that U Π V = 0. If a = b, then Uf(U, f ) Π il9 (U, g) = 0
if U is an open connected neighborhood of a (by virtue of the identity principle, if
fZQ = gzo for some zo € U, then f = g and therefore f = g).
Let X be a set. Suppose that for each χ e X we are given a non-empty family B(x) of
subsets of X such that
• χ e U for any U e B(x),
• for any y e U e B(x) there exists a V e B(y) such that V C U,
• for any U, V e B(x) there exists a W e B(x) such that W C U η V.
The above system (B(x))x e x induces a topology Τ on X: we say that a set Ω c X is open (Ω e T)
if for any χ e Ω there exists a U € B(x) such that U C Ω. Notice that B(x) c Τ, χ e X. Thus the
system (B(x))X G x is a basis of open neighborhoods in T.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-36-320.jpg)
![1.3 Examples of Riemann regions 23
Observe t h a ^ is continuous and π is injective on any neighborhood 11 = ilf (U, / ) .
Moreover, π|n: U —• U is homeomorphic and Or|u)_1
(z) = (Λ, ζ), ζ e U.
Thus (Θ,π) is a Riemann region over C". It is called the sheaf of germs of
holomorphic functions in C".
Notice that
d0((f,a))=d(Taf), f e 0 a , a e Cn
.
Let γ: [0, 1] —• C" be a curve. Suppose that for each t e [0, 1] we are given a
convergent power series σ (t) with the center at γ (t) (or, equivalently, a germ o(t ) from
0K(,)) such that the mapping [0,1] 3 t —> y(t) := (σ(ΐ), y(t)) € 0 is continuous.
Then we say that the series σ(0) can be continued analytically along the curve γ (to
the series σ()). In other words, the series σ(0) can be continued analytically along
γ if there exists a lifting γ : [0, 1] —• Θ of γ such that y(0) = (σ(0), y(0)).
Remark 1.3.2. (a)DefineF: 0 —• C,F((f,z)) := f(z). Then for any it = ilf (i/, / )
we get F ο (ττ |u)_1
= / on U. This shows that F is holomorphic on Θ (F e 0(0)).
(b) For any open set Ω c C and for any / e 0(Ω) define s/: Ω —• Θ,
A
Sf(z) := ( f z , ζ), ζ € Ω. It is clear that sj is a section of (0, ττ) over Ω. Conversely,
if s: Ω —• 0 is a section, then the function / := F ο s is holomorphic on Ω and
s = sf. Thus, Γ(Ω, (0, ττ)) ~ 0(Ω).
(c) Let G C C" be a domain, let / e 0(G), and let / denote the connected
component of 0 that contains Sf(G). Suppose that for some a e G there exists
a curve γ: [0, 1] —• C" such that }/(0) = a and that the series σ(0) := Taf
can be continued along γ to a series σ(1). Since γ: [0, 1] —• 0 is a curve and
p(0) g Sf(G), we conclude that y([0, 1]) c / , which shows that σ(1) e / .
(d) Let dv
: 0 —• 0, 9v
(f, a) := (dv
f, a). Then dv
is continuous and π ο dv
=
π. In particular, 3V
is locally biholomorphic (and, consequently, open). Note that
9υ
(5/ (Ω))=59 ,/ (Ω).
(e) Let η = 1 and let Λ C 0 be a domain. Define
:={[(V,g)h: 3[(u,f)]atA: f : U ^ V is bijective , f(a) = b, g = f~1
}.
One can easily prove that C 0 is also a domain (Exercise).
In particular, if / : G —> D is biholomorphic (D, G C C), then (s/(G))_1
=
sf-i(D).
Example 1.3.3 (Riemann surface of /"1
). Let G C C be a domain and let / : G —• D
be a locally biholomorphic mapping, D := f(G).
Evidently, (G, f ) is a Riemann domain over C. Moreover, ((s/(G))-1
, π) is a
Riemann domain over C.
Let X := {(u;, ζ) e G χ C: f(w) = z}. We endow X with the topology induced
from C2
. Let p(w, z) :— z, (w, z) e X. Then, by the implicit function theorem,
ρ: X — C is locally homeomorphic. Thus (X, p) is a Riemann domain over C.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-37-320.jpg)
![24 1 Riemann domains
We say that Riemann domains (Χι, p), (X2, pi) are isomorphic if there exists a
homeomorphic mapping a: X — • X2 such that p2 ο a — p; cf. § 1.4.
We claim that the Riemann domains (G, / ) , (X, p), and {{Sf{G))~x
, π) are iso-
morphic.
Let φ: G —» Χ, φ(νυ) := (w, f(w)). Then φ: G —» X is homeomorphic
ζ) — w) and ρ ο φ = f . This shows that (G, / ) and (X, p) are isomorphic.
Now let ψ: X — • (sf(G))~l
, yjr{w, z) (gz, z), where g is a local inverse to /
such that g(z) = w. It is clear that ψ is a homeomorphism (ψ~ι
(§ζ , ζ) = (g(z), ζ))
and π οψ = p. Thus (Χ, ρ) and ((s/(G))_ 1
, π) are also isomorphic.
Observe that the function s := φ~1
is holomorphic on X and / ο s = p. The
domain (X, p) is called the Riemann surface of / - 1
.
We would like to mention the following two particular cases.
1°: f ( w ) = wk
,w€G := C* (k g N).
We already know that the domains (X, p) and (C*, / ) are isomorphic. Neverthe-
less, since ρ coincides with the standard projection, the domain (X, p) has a more
geometric nature.
For any point zo € C* the stalk p~l
(zo) consists of exactly k points, say Ao,...,
Ak-1, Aj = (woj, zo), j = 0 , . . . , k - 1. Note that {wo,o, · · ·, m,k-i} = V^ö- Let
U C C* be any simply connected open neighborhood of zo and let fo, ... ,fk-i e
0(U) be holomorphic branches of the k-th root such that fj(zo) = WQJ, j =
0 , . . . , k - 1. Then for each j = 0 , . . . , it - 1 the set Uj := {(/;·(ζ), ζ): ζ e U}
is a univalent neighborhood of Aj and (ρυ})~1
(ζ) = ( f j ( z ) , z ) , ζ e U. In par-
ticular, let zo := 1
> wo,j '•= exp(2nij/k), j = 0 , . . . ,k — 1, U := C Let
y(t) = φ(οχρ(2πίΐ)), t e [0, 1] (recall that^(w;) = (w, wk
)). Then γ: [0, 1] —> X
is a curve such that Y(j/k) = Aj, j = 0 , . . . , k — 1, y(l) = y(0) = Ao- It is clear
that y(t) 6 Uj for t e 7), where T0 := [0, U l] and Tj :=
j = l , . . . , k - l .
This leads to the following geometric interpretation of the domain X. We take k
copies of the domain C say Co,..., Cjt_i. Let ζ <
£ Cj be such that Re ζ < 0 .
We say that ζ lies on the 'upper part of Cj' if Im ζ > 0. Otherwise, we say that ζ lies
on the 'lower part of Cj'. Now we glue the domains Co,..., Ck- together, crosswise
along R_. That is, first we glue Co to C in such a way that we join the upper part of
Co with the lower part on C1, next we glue the upper part of C to the lower of C2,
and so on. Finally, we glue the upper part of Ck- to the lower part of Co-
One can prove that the above construction is impossible in R3
without self inter-
sections.
2°: f(w) := exp(u;), w e G := C.
The geometric interpretation of (X, p) is following. We take a countable family
(Cj)jez of copies of C Next, for each j e Z, we glue Cj to Cj+ crosswise
along M_, joining the upper part of Cj with the lower part of Cj+1. Observe that this
construction can be done in R3
.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-38-320.jpg)
![26 1 Riemann domains
Sometimes, to simplify notation, we will write
/φ
:=(<p*)-l
(f), fe<p*(0(Y)).
(h) φ* ο = dv
x ο ψ*, ν e Ζ. In particular,
Τφ(χ)/ = Tx(f ο φ), f e (9(F), χ eX.
Indeed, using (e) we get pa = qV(a) ° φ and hence, for g e Θ(Υ) and a e X, we
have:
dv
Yg(<p(a)) = dv
(g ο q~(a))(q(<p(a)) = dv
((g ο φ) ο ρ~Χ
)(ρ{α)) = dv
x{g ο φ)(α).
(i) In view of (h) we get d(Taf) > dY(<p(a)) for any a e X and / € φ*(Θ{Υ)).
Now we are in a position to define the fundamental notion of holomorphic ex-
tendibility in the category of Riemann domains over Cn
.
Definition 1.4.3. Let (X, p) be a Riemann domain over C" and let 0 Φ S C <9(X).
We say that a morphism a : (X, p) —> (F, q) is an £-extension if 4 C a*(0(Y))
(i.e. for each / e S there exists age Θ(Υ) such that g ο or = /).
If S = Θ(Χ), then we say that α: (Χ, ρ) —> (Y, q) is a holomorphic extension.
Obviously, if α: (Χ, ρ) —• (Ύ, q) is an ^-extension, then it is an ^'-extension
for any 0 Φ S' C S. Consequently, the most important is the case S = &(X).
The above definition of the S-extension generalizes in a natural way to the case of
Riemann regions over C" but in this case, to avoid situations where the region (F, q)
is 'too large', we assume that every connected component of F intersects φ(Χ) (which
is equivalent to the injectivity of the mapping φ*).
We say that a family 0 φ S c 0(X) is d-stable if dv
xf e S for any / e S and
for any ν e Z" (equivalently, ^ e $ for any / e S and j = ... ,n).
Remark 1.4.4. Observe that for any family S c Θ(Χ), the smallest 3-stable subal-
gebra of Θ(Χ) with 4 U {pi,..., pn) c [4] may be described in the following
way17)
:
[S]:={P(fi,...,fN): Ν eN, Ρ e J>(CN
), 18)
where
*'--={dv
xf: f €<SU{Pu...,Pn}, veZn
+}.
17
> Notice that 1 e [4].
denotes the space of all complex polynomials of Ν complex variables.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-40-320.jpg)
![1.5 The boundary of a Riemann domain 29
1.5 The boundary of a Riemann domain
Let (Χ, ρ), (Y, q) be Riemann domains over Cn and let φ: X — > Y be a morphism.
=φ
Our aim is to define an abstract boundary 3 X of X w.r.t. the morphism φ. The idea
of such an abstract boundary is due to Grauert (cf. [Oka 1984], [Gra 1956], [Gra-Rem
1957], [Gra-Rem 1956], [Doc-Gra I960]).
In the case where (X, p) = (G, id) (G is a domain in Cn ), (Y, q) = ( C , id),
=id
φ = id, the abstract boundary d G will coincide with the set of, so-called, prime ends
of G.
At first let us recall some facts and notions from topology.
Let X be a topological space. We say that a non-empty family $ of subsets of X
is a filter if:
• Αι, a2 € y = » Αι η A2 G
A non-empty family φ of non-empty subsets of X is said to be a. filter basis if:
• VA1,A2e«p : A C Αι (Ί A2-
It is clear that for each filter basis the family := {A c X : 3ße<p: Β c A}
is a filter.
We say that a filter # is convergent to a point a G X if each neighborhood of a
belongs to We shortly write a G lim J.
We say that a filter basis φ is convergent to α if α e lim (equivalently, each
neighborhood of a contains an element of φ); we put lim := lim
We say that a is an accumulation point of a filter $ (resp. filter basis φ ) if a G Ä
for any A € # (resp. A G φ).
Let us recall a few elementary properties of filters (cf. [Bou 1965]):
• If # C & are filters and if a is an accumulation point of then α is an
accumulation point of 5·
• If a G lim & then a e lim for any filter D
• If a is an accumulation point of J, then there exists a filter D $ such that
a G LIMY'.
• a G Ä iff there exists a filter basis φ consisting of subsets of A such that
a G lim φ.
• Let Y be another topological space and let φ: X — • Y. Then φ is continuous
iff for any filter basis φ in X the filter basis :— {φ(Α): A G φ } satisfies the
relation: ^(Ιίπιφ) c Ιίπι^(φ).
• X is Hausdorff iff any filter in X converges to at most one point. If X is a
Hausdorff space and lim # = {«}, then we write lim % = a.
For a G X and A C X let Q5C(A, A) denote the family of all open connected
neighborhoods U of a with U C A. Put <8C (A) : = <8C(A, X ) .
Let us come back to the situation when (Χ, ρ), (Y, q) are Riemann domains over
C" and φ: X — > Y is a morphism.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-43-320.jpg)
![30 1 Riemann domains
Definition 1.5.1. We say that a filter basis α of subdomains of X is a φ-boundary
point of X if:
(1) α has no accumulation points in X,
(2) there exists a point yo e Y such that lim φ(a) = yo,
(3) for any V e Q3c(_yo) there exists exactly one connected component U =:
e(a, V) of (p~HV) such that U e a,
(4) for any U e α there exists a V e Q3c(jo) such that U = 6(α, V).
=<p
Let 3 X denote the set of all ^-boundary points of X. We put
=φ =φ
Χ := XU dx
and we extend φ to φ: X —> Y by putting φ(a) := yo if a and yo are as above.
=<p
Moreover, w e put ρ : = q ο φ.
In the special case when (Y,q) = (Cn
, id) and φ = ρ we skip the superscript p
= = =p =p
and we write dX and X instead of 3 X and X , respectively.
Note that:
• If (Ύ, q) = (X, p) and φ = idx, then the «^-boundary is empty.
• If G = E[ 0, 1), then 3 G consists of points from (3G)(0, 1] with'multiplicity'
1 and of points from (0, 1] with 'multiplicity' 2.
Remark 1.5.2. If α satisfies (2), (3), and (4) and α has an accumulation point ;to € X,
then lim α = *ο·
Indeed, first observe that yo = lim<p(a) = φ(χο). Let U e 93c(*o) such that
φυ is injective. Observe that U is a connected component of φ'~φ{ϋ))19)
. Take
V e 23c(yo) with V <ξ <p(U). We know that jc0 e V) =: A. Since A is
connected, φ(Α) C <p(U), and ^o € A, we conclude that A c U. Thus U e
which finishes the proof.
Instead of filters, one can describe the -boundary using sequences of curves:
Consider the family β of all sequences £ = C X such that:
(a) (xk)kLi has no accumulation points in X,
(b) there exists a yo ^ Y such that l i m ^ + 0 0 (p(xk) = yo>
(c) for any neighborhood V of yo there exists a ko such that for any k, I > ko there
exists a curve γ^ι: [0, 1] — > <p~l
{V) such that η,ι(0) = Xk, n,/(l) = xi•
Lemma 1.5.3. Let y = C X, (tk)kLi C (0,1), tk < tk+ι, tk —>· 1. Then the
following conditions are equivalent:
(i) (xk)fLi satisfies (b) and (c);
We argue as in Remark 1.1.2(f): Let Uq be a connected component of (<p{U)) that
contains U. Then ψ := ° <PuQ maps Uq into U and ψ — id on U. Consequently, by the
identity principle, ψ — id on Uq, i.e. Uq = U.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-44-320.jpg)
![1.5 The boundary of a Riemann domain 31
(ii) there exists a curve γ: [0, 1) —• X such that y(tk) = Xk, k > 1, and the limit
l i m ^ i φ ο γ (t) exists 20
Proof, (ii) =>• (i): Clearly, lim*->.+«>?(**) = lim*^+ 0 0 <p(y(tk)) =: yo- Let V be
an arbitrary neighborhood of yo- Take ko 1 such that γ it) e V for t > t^. Then
for any i > k >kolhe curve y l^,^] connects jc^ with χι in <p~l
(V).
(i) =>· (ii): Let ( V / ) ^ be a basis of neighborhoods of yo with V/+i C Vj, j > 1.
For any j there exists a k(j) such that for any k > k(j) the points Xk, Xk+ can be
connected in (p~l
(Vj) by a curve (we may assume that k(j) < k(j + 1)). Let yj be
a curve connecting the points xk(j), · · ·, Xk(j+) in ^c1
(Vy), j > 1, and let yo be an
arbitrary curve connecting χ ι , . . . , xjt(i) in X. Now, we glue the curves step by step
yo U γι U Y2 U . . . and, after an appropriate change of parametrization, we get a curve
γ: [0, 1) —> X with γ(t^) = Xk, k > 1. It is clear that limf_+i φ °9(t) = yo· Π
Remark 1.5.4. Let j: = C X be a sequence such that there exist [0, 1) 3
tk / 1 and a curve γ : [0, 1) —> X with the following properties:
• xk = y(tk),k > 1,
• dx(xk) —> 0,
• y0 •'= lim*-*.+oo <p{xk) exists,
φ the limit limf ^i ρ ο y (?) = zo e C" exists 2 l
Then y 6 Θ.
Indeed, by Lemma 1.5.3, it suffices to check that l i m ^ i φ ο γ (ί) = yo· Observe
thatgo^oy = ρογ. Hence q(yo) — zo- Let V be an arbitrary univalent neighborhood
of yo- Take ko such that (p(xk) e V for k > ko and ρ ο γ (ί) e q(V) for t > to := tk0.
Consider the curves
σι :— φ ο y |[>0,i): [ίο, 1) —• Υ,
σ2 := (qvrl
(p°Y[to,i))·· Ho, Ό —• V.
Observe that σι (ίο) = &2(to) = <p(xk0) and Ι οσ
ΐ = q οσ2 = ρ ο y |[ί0,ΐ). Hence, by
the identity principle for liftings, σ = σ2. In particular, φ ο y (ί) g V for t > ίο-
Remark 1.5.5. If (xk)^Lι satisfies (b) and (c) and has an accumulation point
xo € X, then lim*_>.+00 Xk = xq (cf. Remark 1.5.2).
Indeed, assume that limi^+oo Xkv — Let U e Q3c(xo) be such that φυ is
injective. Note that U is a connected component of φ~λ
(φ(ί/)). Let ko be such that
for any k, I > ko the points λ:*, χι can be joint in φ~ι
(φ(ϋ)) by a curve y*^. Now
let v
>
o be such that kvo > ko and Xkv e U for ν > ν>ο· In particular, any point Xk with
k > ko can be joint with x^ in φ~χ
{φ{υ)). Hence Xk e U, k > fco·
For £ = (xk)™=v l' = (x'k)?=l e Θ we write y ~ p' whenever:
Equivalently: φ ο γ extends to a curve γ: [0, 1] —> Υ.
21
')
Equivalently: the curve ρ ο γ : [0, 1) —> C" extends continuously to [0, 1],](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-45-320.jpg)
![32 1 Riemann domains
• lim^+oo <p(xk) = linu_»+00 <p(x'k) =:
• for any neighborhood V of yo there exists a ko such that for any k > ko the points
Xk and x'k can be connected in <p~x
(V) by a curve.
Note that y is equivalent to each of its subsequences. Observe that is an
equivalence relation. Put
dX := Θ/~
SS ^ ψ SS
and define φ : 3 Χ —> Υ, φ ([y]~) := lim^+oo <p(xk)·
=φ χζφ
Proposition 1.5.6. There exists a canonical bijection Ξ: 3 X —> 3 X such that
« =<p
φ ο Ξ = φ on 3 Χ.
=φ
Proof. Let a € d Χ. Let ( V * j be a basis of connected open neighborhoods of yo :=
<p(a) with Vjt+i C Vk, k > 1. For each k take an arbitrary point Xk e £4 := C(a, Vk)
and let ρ := (xk)fLi· Obviously, 1ΐηι^+ 0 0 <p(xk) = yo- Observe that Uk+ι C Uk
and, therefore, Xk can be connected with xg in <p~l
(Vk0) for any k, I > ko. Thus y
satisfies conditions (b) and (c) (from the definition of the family β). Suppose that
(x
k)kL has an accumulation point jco in X. Take an arbitrary U = β(α, V) e ο and
let Vk0 C V. Then Uk0 C U and hence Xk € U for k > ko. In particular, xq e Ü.
Thus jco is an accumulation point of a; contradiction. We have proved that y € 6 .
Now let (VpJ^j be another basis of neighborhoods of yo and let y' = C ^ ) ^ be
constructed with respect to {Vk)^=l (x'k e U'k := C(a, V£), k > 1). We will show
that y ~ y'. Take an arbitrary neighborhood V g ®c(>o) and let Vk0 U V^ c V.
Then Uk0 U U'k(j c C(o, V) and, consequently, for any k > ko the points Xk, x'k can be
connected in β(α, V) C <p~l
{V).
=φ πφ
Thus we can define a mapping Ξ: dX —> 3Χ, Ξ(α) := [y]~. Obviously
« = =<p
φ ο Ξ = φ on 3 Χ.
We will prove that Ξ is injective. Suppose that Ξ (α) — Ξ(α'). In particular,
φ(ο) = φ{α!) —. yo- Fix a basis (Vfc)^i of connected neighborhoods of yo with
Vk+i C Vk,k> 1, and let xk e Uk := 6(α, V*), x'k € U'k := 6(α', Vk), k > 1,
y := (xk)kLi* := W e
know t h a t
? ~ ?'· T a k e
an arbitrary U = e(a, V)
and let ko be such that for any k > ko the points xk, x'k can be connected in (p~x
{V).
Let k > ko be such that Vk C V. Then Uk U U'k c U. Hence 6(α', V) = U and
therefore U e a'. Thus α = α'.
It remains to show that Ξ is suijective. Let y = (xk)kLi e β . Observe that for
any connected neighborhood V of yo := linu^+oo (p(*k) there exists exactly one
connected component, say Gy, of <p~l
{V) such that ^ e Gv for fc » 1. It clear
that Gy remains the same if we substitute y by an equivalent sequence. Obviously, if
W C V, then Gw C Gv. Put o: = {Gv: V e ®c(yo)}· It is clear that α is a filter
basis and that lim φ(a) = yo· We will show that ο has no accumulation points in X.
Suppose that xq is an accumulation point of o. Then, by Remark 1.5.2, lim α = χο· In](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-46-320.jpg)
![38 1 Riemann domains
• W w r k v ^ p j c w .
Observe that W U Σ] is an open subset of Χ, Σι c W, and that int Σι = 0 . Put
ψ := <pw. For y e P0 := Ρ let
a(y) := { V ( V ) : V' G V)},
where V ( V ) denotes the connected component of φ~]
(V') that contains the set
( V'). Notice that PQ is a relatively closed subset of P.
Step 1°. a(z) e dX, ζ e P0:
Fix a yo G PQ. It is clear that a(yo) is a filter basis satisfying conditions (2), (3),
(4) of Definition 1.5.1 (lim qp(a(;yo)) = Jo)· Suppose that a(yo) has an accumulation
point xo G X. Obviously, φ(χο) = yo· Observe that G W.
Otherwise, we would find W,W2 e Q3c(jc0), V,V2 G ®cOo, V) such that
V2 <£ V, Wi Π W = 0 , : Wi —> V is biholomoφhic, and = V2. Put
U2 := V(V2). Observe that W3 := f ~ V 2 P) C U2. Since W2 Π U2 φ 0 (jcq is
an accumulation point of a(;yo)), we get W2 c U2. Fix an χι e W3 and connect this
point with xo inside U2 by a curve γ. Then φ ο γ lies inside V2 although γ crosses
W2 contradiction.
Now choose W' e V' e 95c(yo, V) such that <pw> W' — • V is
biholomorphic. We will show that (φν')~Χ
(ν>
P ) <ZW. Fix an x' e W' Π W and
suppose there exists an x" e W' W with φ{χ") G V' P. Connect x' with x" by
a curve γ inside W' (φψ')~Χ
(Υ' Π Ρ). Then φ ο γ is a curve inside V P. So
ο {φ ο γ) is well defined. By the identity principle for liftings both curves are
equal. In particular, χ" = γ (1) g W contradiction.
This shows that jco € Σι; contradiction.
Define
U := WU Σι U {α(ζ) : ζ e P0}·
We claim that U satisfies all the required conditions.
Step 2°. φ: U —> V is bijective:
The suijectivity is clear.
Take x', x" e U, χ' ψ χ " , such that ψ ( χ ' ) = <p{x") =: yo· The only case which
is not obvious is that when χ', x" g Σι . Then there are disjoint neighborhoods W' e
y$c{x') and W" e ^ d x " ) and V' e ®c(>>o, V) such that φ maps biholomorphically
W onto V' and W" onto V'. Take a point y G V' P. Then y has two preimages in
W; contradiction.
*φ
Step 3°. U is an open subset of X :
*<p
First observe that U c X • Indeed, the set PQ as a relatively closed subset of Ρ is
=<p
thin and the mapping Φ: U dX —• V PQ is biholomorphic.
=<p
Thus, it suffices to show that U is an open subset of X .](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-52-320.jpg)
![1.6 Union, intersection, and direct limit of Riemann domains 39
Let a(jo) be one of our filter basis with yo e Po. Choose V, V2 G ®c(>o, V),
V2 V], and let U :— V(Vi). Take as a neighborhood of a(jo) the following set
u '·= Π (Ü)a(y0)·
We will prove that U C U. First we prove that i / i C W U E , = : W.
Otherwise, we can find points x' G U Π W, x" G U W. Let γ be a curve in
U1 connecting x' and x". Let ίο be the first parameter such that y (ίο) G 3 W. Put
x* := y(to). Then χ* £ Σι. It is clear that χ* G 3W. Observe that φ ο γ is a curve
in V starting at the point φ{χ') G V P.
Suppose that y* := φ{χ*) £ P. Choose W3 G ®c(jc*) and V3 e V P)
such that <p|w3: — • V3 is biholomoφhic. Then, since W3 Π W Φ 0 , we get
W3 C W (use the identity principle for liftings); contradiction.
So we know that y* e Ρ Π V. Now, as in Step 1°, we show that x* e Σι;
contradiction.
Hence we know that Ui c W.
=<p
Take α e U Π 3 X. We want to show that α e U. Put y' := φ{α). Of course, then
/ e V2 <£ V. Observe that y' e Ρ Π V2.
For, otherwise y' e V P. Hence there exists exactly one point w' e W with
φ{ν') = y'. We claim that in such a case w' must be an accumulation point of a.
For, fix a U' e a. Let W' e W) and U" = e(o, <p{W')). Observe that if
U" Π W φ 0 , then U" = W'. Recall that U e j f l . We already know that U C W.
Consequently, U" Π W Φ 0 . Since int Σι = 0, we get U" Π W Φ 0 and hence
U" = W'. In particular, U'nW' φ 0 .
Assuming that yr
€ φ(Σ,) also leads to a contradiction.
So we start with the information that <p(A) =: y' € Pq. We claim now that
α = α(/).
Otherwise there are U' e α and U" = V( V") e a(y') with U' η U" = 0 . We may
assume (cf. property (*) from the proof of Proposition 1.5.11) that U' = C(o, V")
and that V" C In particular, U' <ZU. Hence U' = U"; contradiction.
The proof of Proposition 1.5.13 is completed. •
1.6 Union, intersection, and direct limit of Riemann domains
The union of Riemann domains. Let (X, p) be a Riemann domain over C" and let
0 φ £ c Θ(Χ). Consider a family α,·: (Χ, ρ) —> (F,, qi), i G I, of -^-extensions.
We would like to define an -ί-extension a : (Χ, ρ) —> (Υ, q) which behaves like the
union of the above extensions (cf. [Gra-Fri 1976], [Ohg 1979]).
Definition 1.6.1. We say that S weakly separates points in X if for any x,x2 G X
with jci Φ JC2 and p{x) = p(x2) there exist f € S and ν G Z" such that 3 ( j c 1) Φ](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-53-320.jpg)
![40 1 Riemann domains
dv
f(x2); equivalently, for any χι,χ2 € X with x φ χι there exists an f e S such
that TXl f φ TX2f (as formal power series 24
^).
Observe that the condition is empty if X is univalent. In particular, if G is a domain
in Cn
, then any family <8 weakly separates points in G.
If S is 3-stable, then the weak separation means that the family -8 separates points
in stalks. Consequently, if -8 is 3-stable and ρ e <$n
, then -8 weakly separates points
in X iff it separates points in X.
We say that an ^-extension α: (X, p) —> (F, q) satisfies condition (S) if the
family <8a
weakly separates points in Y.
Proposition 1.6.2. There exists an -8-extension a: (Χ, ρ) —> (y, q) satisfying (S)
such that:
• for any i e I there exists a morphism <pt: (7,·, qi) —> (7, q) with φι ο αι = a,
•U ieJ<Pi(Yi) = r,
• if β: (Χ, ρ) —> (Ζ, r) is an -8-extension satisfying (S) such that there exist
morphisms ψι: (7( , qi) —> (Z, r), i e I, with ψι ο α,· — β, i e I, then there exists a
morphism σ : (Κ, q) —> (Ζ, r) such that σ οφι = ψί, i e I (inparticular, σοα = β).
Moreover, the extension a: (X, p) —> (Y, q) satisfying the above conditions
is uniquely determined up to an isomorphism.
Observe that the morphisms φι: (Yi,qi) —• (Y,q), i € /, are uniquely deter-
mined.
We write
(a: (X, p) —-> (Y, q)) = J(ai: (X, p) —• (Yi,qi))
iel
and we say that a: (X, p) —> (F, q) is the union of the family
cn:(x, p)—>(Yi,qi), i e I.
Proof For any i e I and / e S, let /,· e &{Yi) be such that /,· ο α,· = / (i.e.
fi := /«<)· Let
f : = | J r / x{i}.
For (ji, i'i), (y2, h) e f we put
(yi.ii) ~ (yi, 12) :«=> Tylfil = Ty2fi2, f e <8;
(in particular, qii (yj) = qi2(yi) and fh (yi) = fi2(yi) for any / e -8). We define
q:Y-+ Cn
, q([(y,i)]~):=qi(y),
f : Y ^ C , f[(y, i)]~) := My), f € 4,
ψί: Yi Y, <pi(y) := [(y, i)]~, i e I.
24
) Recall that two formal power series Σν<ζζ"+ av(z — a)v
, bv(z — b)v
are equal if
a = b and av - bv for any ν e Z!j_.](https://image.slidesharecdn.com/25561862-250514164531-64f251b4/85/Extension-Of-Holomorphic-Functions-Marek-Jarnicki-Peter-Pflug-54-320.jpg)
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Extension Of Holomorphic Functions Marek Jarnicki Peter Pflug
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- 5. de Gruyter Expositions in Mathematics 34 Editors Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R.O.Wells, Jr., Rice University, Houston
- 6. de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) 2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues 3 The Stefan Problem, A. M. Meirmanov 4 Finite Soluble Groups, K. Doerk, T. O. Hawkes 5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin 6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin 7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, M. V.Zaicev 8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro 9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug 10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini 11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao 12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub 13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, 5. A. Nazarov, B. A. Plamenevsky 14 Subgroup Lattices of Groups, R Schmidt 15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H. Tiep 16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig 19 Blow-up in Quasilinear Parabolic Equations, A. A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov 20 Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D. Lawson, Ε. B. Vinberg (Eds.) 21 Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöfer, Η. Hähl, R. Löwen, M. Stroppel 22 An Introduction to Lorentz Surfaces, Τ. Weinstein 23 Lectures in Real Geometry, F. Broglia (Ed.) 24 Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V. Pukhnachov, S. I. Shmarev 25 Character Theory of Finite Groups, B. Huppert 26 Positivity in Lie Theory: Open Problems, J. Hilgert, J. D. Lawson, K.-H. Neeb, Ε. B. Vinberg (Eds.) 27 Algebra in the Stone-Cech Compactification, N. Hindman, D. Strauss 28 Holomorphy and Convexity in Lie Theory, K.-H. Neeb 29 Monoids, Acts and Categories, M. Kilp, U. Knauer, Α. V. Mikhalev 30 Relative Homological Algebra, Edgar E. Enochs, Overtoun M. G. Jenda 31 Nonlinear Wave Equations Perturbed by Viscous Terms, Viktor P. Maslov, Petr P. Mosolov 32 Conformal Geometry of Discrete Groups and Manifolds, Boris N. Apanasov 33 Compositions of Quadratic Forms, Daniel B. Shapiro
- 7. Extension of Holomorphic Functions by Marek Jarnicki Peter Pflug W DE _G Walter de Gruyter · Berlin · New York 2000
- 8. Authors Marek Jarnicki Institute of Mathematics Jagiellonian University u. Reymonta 4 30-059 Krakow Poland jarnicki@im.uj.edu.pf Peter Pflug Department of Mathematics Carl von Ossietzky University Oldenburg 26111 Oldenburg Germany pflug@mathematik.uni-oldenburg.de Mathematics Subject Classification 2000: 32-02; 32Axx, 32Dxx, 32Exx, 32Txx, 32Uxx, 32Wxx Key words: Riemann domains, Holomorphic extension, Holomorph convexity, Riemann-Stein domain, Plurisubharmonic function, Pseudoconvexity, Levi problem, Envelope of holomorphy © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress — Cataloging-in-Publication Data Jarnicki, Marek. Extension of holomorphic functions / by Marek Jarnicki, Peter Pflug. p. cm. — (De Gruyter expositions in mathematics ; 34) Includes bibliographical references and index. ISBN 3-11-015363-7 (alk. paper) 1. Holomorphic functions. I. Pflug, Peter, 1943— II. Title. III. Series. QA331 .J37 2000 515'.98-dc21 00-060145 Die Deutsche Bibliothek — Cataloging-in-Publication Data Jarnicki, Marek: Extension of holomorphic functions / by Marek Jarnicki ; Peter Pflug. - Berlin ; New York : de Gruyter, 2000 (De Gruyter expositions in mathematics ; 34) ISBN 3-11-015363-7 © Copyright 2000 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the author's TgX files: I. Zimmermann, Freiburg. Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer-GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.
- 9. To Mariola and Rosel
- 11. Preface Starting from the discussion of the holomorphic logarithm in the plane, it is necessary to introduce Riemann surfaces in order to be able to deal with the maximal domain of existence of that function, but without being bothered about its possible multival- uedness. On the other hand, any domain in the complex plane is the existence domain of at least one holomorphic function, i.e. there exists a holomorphic function that is not the restriction of another holomorphic function having a strictly larger domain of definition. In the n-dimensional situation (n > 2) it was already observed by Hartogs that there are pairs of domains G c G', G φ G', such that any holomorphic function on G extends holomorphically to G'. Even more, simultaneously one has to handle the problem that the extended functions may be multivalued. This phenomenon has led to the notion of Riemann domains over C". Exactly this category of objects and the theory of holomorphic functions on them is the subject of our book. We try to give a systematic representation of domains of holomorphy and envelopes of holomorphy in that category. The authors feel that a lot of results they are presenting have never been published in book form. We are not touching the theory of holomorphic functions on complex spaces or even on complex manifolds. Instead, we continue investigating domains of holomorphy for special classes of holomorphic functions on special types of domains. Our interest in this area of complex analysis started directly after our studies when both of us were working on the continuation of holomorphic functions. Although we had changed our fields of interest, we were attracted by such questions all the time, and we were following the development in that direction. During the years we got the impression that there is a need of a source where the main results are collected. We hope this book can serve as such a source. The choice of topics obviously reflects our personal preferences. For example, we will solve the Levi problem via the 3-problem and functions of restricted growth. Our idea is to address this book to everybody who likes to extend her/his knowl- edge beyond the standard course in several complex variables in Crt . We tried to make the book as complete as possible and to keep the results used without proving them as limited as possible. Nevertheless, for some parts we will have to use certain facts (for example consequences of Theorems A and B) without giving proofs. The same will happen also with some facts on plurisubharmonic functions (related to the Monge-Ampere operator), although we present an extended section on plurisubhar- monic functions. Textbooks that support the reader can be found in the first part of the bibliography.
- 12. viii Preface We should point out that the bibliography is far from being complete. We included only the papers that we had studied during the preparation of the book or before. So we have to apologize if readers are interested in historical developments of presented results. The theory of extension of holomorphic functions contains a lot of questions which are still waiting for being solved. We have put many of them into the text (marking them by [T]). The reader is encouraged to work on some of them. During the process of proofreading we got the impression that this became a never- ending story and there had to be a time for us to stop. We would be pleased if the reader would inform us of any errors he/she may have detected while studying the text. It is our deep pleasure to thank our teachers, Professors H. Grauert and J. Siciak, who taught us the beauty of complex analysis. We would also like to thank Dr. habil. W. Zwonek who helped us in corrections of the text. We thank the following institutions: Committee of Scientific Research (KBN), Warsaw (PB 2 P03A 060 08, PB 2 P03A 017 14), Volkswagen Stiftung (Az. 1/71 062, RiP-program at Oberwolfach), and Niedersächsisches Ministerium für Wissenschaft und Kunst (Az. 15.3-50 113(55) PL). Without their financial support this work would have never been possible. We would also like to thank our universities for support during the preparation of the book. Finally we thank Walter de Gruyter Publishers, especially Dr. M. Karbe, for having encouraged us to write this book. Krakow — Oldenburg, May 2000 Marek Jarnicki Peter Pflug
- 13. Contents Preface vii Chapter 1 Riemann domains 1 1.1 Riemann domains over C" 3 1.2 Holomorphic functions 18 1.3 Examples of Riemann regions 21 1.4 Holomorphic extension of Riemann domains 25 1.5 The boundary of a Riemann domain 29 1.6 Union, intersection, and direct limit of Riemann domains 39 1.7 Domains of existence 50 1.8 Maximal holomorphic extensions 54 1.9 Liftings of holomorphic mappings 1 62 1.10 Holomorphic convexity 75 1.11 Riemann surfaces 86 Chapter 2 Pseudoconvexity 96 2.1 Plurisubharmonic functions 96 2.2 Pseudoconvexity 129 2.3 The Kiselman minimum principle 153 2.4 ä-operator 159 2.5 Solution of the Levi Problem 177 2.6 Regular solutions 184 2.7 Approximation 190 2.8 The Remmert embedding theorem 195 2.9 The Docquier-Grauert criteria 201 2.10 The division theorem 208 2.11 Spectrum 219 2.12 Liftings of holomorphic mappings II 224 Chapter 3 Envelopes of holomorphy for special domains 235 3.1 Univalent envelopes of holomorphy 235 3.2 ^-tubular domains 258
- 14. χ Contents 3.3 Matrix Reinhardt domains 284 3.4 The envelope of holomorphy of Χ Μ 302 3.5 Separately holomorphic functions 324 3.6 Extension of meromorphic functions 334 Chapter 4 Existence domains of special families of holomorphic functions 341 4.1 Special domains 341 4.2 The Ohsawa-Takegoshi extension theorem 388 4.3 The Skoda division theorem 410 4.4 The Catlin-Hakim-Sibony theorem 422 4.5 Structure of envelopes of holomorphy 441 List of symbols 461 Bibliography 469 Index 483
- 15. Chapter 1 Riemann domains Roughly speaking, a Riemann domain over Cn is a 'domain spread over Cn i.e. a connected complex manifold X having a global projection ρ: X —> C" such that ρ is locally homeomorphic. The class of Riemann domains over Cn extends the class of subdomains of Cn . Riemann domains appear in a very natural way while discussing problems related to holomorphic continuation. Consider the following classical example. Let G := C and let Log denote the principal branch of the logarithm. Then the function Log extends holomorphically to a domain X which is no longer a plane domain but a so-called Riemann surface of the function Log. The same phenomenon appears if we consider on G a branch of the k-th root. This means that the maximal domain of existence of a holomorphic function defined in a plane domain may be non-univalent — it may be a Riemann domain over C. More generally, given a domain G c C", we can consider the maximal domain of existence of a family 4 c 0(G), where 0(G) denotes the space of all functions holomorphic on G. The most interesting case is the case when S = 0(G). From the point of view of the theory of holomorphic functions the following three fundamental questions are the most important: (1) Does the maximal domain of existence always exist in the category of Riemann domains over Cn ? (2) What is a characterization of those domains which cannot be holomorphically extended to any larger Riemann domain ? (3) When is the maximal domain of existence univalent, i.e. can it be realized as a domain in C" ? It is well known that an arbitrary domain G C C is the maximal domain of existence of the whole space 0(G), i.e. each boundary point of G is a singular point for a function holomorphic in G. This is no longer true in C" with η >2. For example, consider the following domain (cf. [Sha 1976]): Let Ε denote the unit disc. Put Ρ := Ε χ (2Ε) c C2 and let G := Ρ (Qi U 02 U S), ^ For the notation used in this book see 'List of symbols'.
- 16. 2 1 Riemann domains where ß i := {(jc + ιθ, w) G Ρ: χ > 0, Μ < 1}, 02 := {(0 + i>, w) e P: y>0, w > 1}, 5 := {(* + ry, w)e Ρ: χ >0, y> 0, |w| = 1}. Define Go := P{(0 + iy,w) eP:y> 0} and notice that GQG Φ 0. Let f E 0(G). Then, for any (Z, W) E Go, the Cauchy integral is independent of r (with < r, 1 < r < 2), and consequently, / e (9(Go). Moreover, using the Cauchy integral formula and the identity principle, we see that / = / on the domain {(* + iy, w) g Ρ: χ < 0 or y < 0} U {(z, w)ePQ2, 1 < |w| < 2}. Hence / is a holomorphic extension of / to Go. Thus the maximal domain of existence of 0(G) must contain Go and Go £ G. Let fo: G —> C be given by the formula Then fo e 0{G) and fo(z, w) = f+(z). Moreover, fora± := (η + ϊη, 1 ± η) e Go (0 < η 1) we have: fo(a+) = —/o(«-) and fo(a+) = fo(a^). Consequently, the domain of existence of 0(G) cannot be univalent. The full description of the envelope of holomorphy of the above domain G will be given in § 4.1 (see also Remark 3.1.12(d)). Let us mention that question (1) has a positive answer (for any family £); cf. Thullen Theorem 1.8.4. Problem (2) is completely solved only in the case % = 0(G) (Chapter 2). In the general case, some characterizations are known for special domains and special families of functions (Chapter 4). Problem (3), even in the case S = 0(G), is far from being solved. It will be discussed for special domains G in Chapter 3. |ιυ| < r, 1 < r < 2, fo(z, w) := f-(z) i f M < l f+(z) if |u>| > 1, where f_(z) := ejLog(-z)j f + ( z ) ._ ejLog(iz)+ij
- 17. 1.1 Riemann domains over C" 3 1.1 Riemann domains over C Definition 1.1.1. A pair (Χ, ρ) is called a Riemann region (spread) over C" if: • X is a topological Hausdorff space, • ρ: X —> Cn is locally homeomorphic. If X is connected, then we say that (X, p) is a Riemann domain over Cn . If (X, p) is a Riemann region over C", then for any connected component C of X the pair (C, pc) is a Riemann domain over Cn . If we replace in the above definitions the space C" by a connected «-dimensional complex manifold Μ 3 then we get the notions of a Riemann region over Μ and Riemann domain over M, respectively. The mapping ρ is called the projection. For ζ € p(X) the set is called the stalk over z. A subset A C X is said to be univalent (or schlicht) ii p A - A —• p(A) is homeomorphic. Observe that if A C X is open or compact, then A is univalent iff PA is injective. Notice that, in general, a set A C X such that PA is injective need not be univalent (Exercise). In the sequel we will sometimes identify X with (X, p) if it is clear from the context what the projection ρ is. Remark 1.1.2. (a) If G c Cn is a domain, then (G, idc) is a Riemann domain over CM . This will be the standard identification of domains in C" with Riemann domains. Of course, any locally homeomorphic mapping ρ: G — • Cn makes from G a Riemann domain over C". For example, (C, exp) is a Riemann domain over C. (b) If (X, p) is aRiemann domain over Cn , then ρ is an open mapping. In particular, the set p(X) is a domain in Cn . For any ζ e p{X) the stalk p~x {z) is discrete. (c) If (X, p) is a Riemann domain over C , then the family (U, pu)u. where U 2 > That is, each point a € X has an open neighborhood U such that p(U) is open in C" and pu U —• p(U) is homeomorphic. 3 ' Recall (cf. [Nar 1968]) that a topological Hausdorff space Μ is called an n-dimensional Ck -manifold (resp. complex manifold) if there exists a family (£/,, φ,· ),·<=/ (called an atlas) such that: • (^i)ie/ is an open covering of M, • φι is a homeomorphic mapping of £/; onto an open subset φι (ί/, ) of R" (resp. C"), • for any i, j e I, if t/, Π Uj Φ 0, then the mapping ψ] ο <pfl is of class Gk (resp. holomorphic) on <pi{Ui Π Uj)here k e Z+ U {oo} U {ω}, where ω real analytic. It is clear that any complex «-dimensional manifold is a 2n-dimensional real analytic manifold. Moreover, any Cl -manifold is an C*-manifold for 0 < k < £ < ω (with oo < ω). Suppose that Ν is another m-dimensional -manifold (resp. complex manifold) with an atlas (Vj, ipj)jej. We say that a continuous mapping F: X —> Y is of class Gk (resp. holomorphic) if • for any i e I, j e J, the mapping ψ} ο F ο φ7ι is of class Gk (resp. holomorphic) on <p, (t/,· Π F"UVj)) in the classical sense; we write: F e Ck (M, N) (resp. F e 0(M, N)). Put Ck (M) := Ck (M, C), Θ{Μ) := Θ(Μ, C). Notice that any connected manifold is arcwise connected.
- 18. 4 1 Riemann domains runs over all univalent open subsets of X, introduces an atlas of an n-dimensional complex manifold on X. Evidently, not all connected complex «-dimensional manifolds are Riemann do- mains; e.g. a compact complex manifold cannot be a Riemann domain. In the category of non-compact complex manifolds the situation is as follows: — if η = 1, then any non-compact connected 1-dimensional complex manifold is a Riemann domain over C (with suitable projection ρ); cf. Theorem 1.11.1, — if η > 2 , then there exist very regular non-compact connected complex mani- folds which are not Riemann domains over C"; cf. Remark 1.11.11. (d) If (X, p) is a Riemann domain over Cn , then (Υ, ργ) is a Riemann domain over CM for any domain Y C X. (e) If (X, p), (F, q) are Riemann domains over Cn and Cm , respectively, then (Χ χ Υ, ρ χ q)4) is a Riemann domain over Cn+m . (f) Let (X, p) be a Riemann domain over Cn and let Y be an open univalent subset such that p(Y) = p(X). Then Υ = X. Indeed, suppose that there exists a point a e dY. Let U be an open univalent neighborhood of a. Put φ := (p|y)- 1 ο (pu)'- U —• Y C X. Then φ(χ) = χ for χ g U Π Y. In particular, by continuity, φ(a) = a; contradiction. Definition 1.1.3. Let (X, p) be a Riemann domain over C", let Γ be a topological space, and let γ: Τ —> Cn be a continuous mapping. Any continuous mapping γ : Τ —>· X such that ρ ο γ = γ is called a lifting of γ to (X, p). If Τ C p(X), then any lifting s of id7- to (X, p) is called a section of(X, p) over T. Let Γ(Τ, (Χ, ρ)) denote the set of all sections of (X, p) over T. Remark 1.1.4. (a) A set A C X is univalent if and only if there exists a section s e Γ(ρ(Α), (Χ, ρ)) such that s(/>(A)) = A. (b) By Remark 1.1.2(f) there are no global sections of (X, p) over p(X) unless X is univalent. Proposition 1.1.5 (Identity principle for liftings). Let Τ be a connected topological space, to € T, and let yj : Τ —>- X, j = 1,2, be two liftings of a continuous mapping γ : Τ —> p(X) such that yi(io) = 72(^0)· Then γ = fo. Proof Let 7o := {t E T : yi(t) = yiit)}', TQ is closed and non-empty. It remains to show that 7o is open. Take a u e TQ. Let a := y (u) = 72(u) and let U be an open univalent neighborhood of a. By continuity there exists an open neighborhood V of Μ such that yj(V) cU,j = 1, 2. Then yjv = (pu)~l ο yv, j = 1,2. Thus V C T0. • Corollary 1.1.6. (a) If a set Τ C p(X) is connected and si, S2 € Γ{Τ, (Χ, ρ)) are such that s (zo) = $2(10) for a point zo £ Τ, then si = S2- 4 > (pxq)(x,y):=(p(x),q(y)).
- 19. 1.1 Riemann domains over C" 5 (b) Let A i, A2 C X be univalent sets such that • ΑχΓΑ2φ0, • the set p(A) Π p(A2) is connected- Then the set Αι U A2 is univalent. Notice that the conclusion of (b) does not hold if A Π A2 = 0 or p ( A ι ) Π p(A2 ) is disconnected. Proof, (a) follows directly from the identity principle. (b) Fix an x0 e At Π A2. Let Τ := p(A) Π p(A2), Sj := {ρΑ])~Χ τ, j = 1, 2. Observe that si and s2 are sections over Τ such that s(p(xo)) — s2(p(xo)). Consequently, by (a), s = s2, which finishes the proof. • Let (X, p) be a Riemann domain over C". Since ρ is locally homeomorphic, we can introduce on X the notion of a ball. More precisely, let q : C" — • M+ be an arbitrary C-norm. Put Bq(zo, r) := {z e Cn : q(z - z0) < r}, z0 e Cn , r e (0, +oo], 5) For a e X and r e (0, +oo] let Bq(a, r) = Bx^(a, r) denote an open univalent neighborhood of a (if it exists) such that p(Bq(a, r)) = Bq(p(a),r). Observe that: • Bq(a, r) exists for small r > 0; • if Bq(a, r) exists, then it is uniquely determined; • if Bq(a, r) exists, then for any a' € Bq(a, r) andO < r' < r — q(p(a') — p(a)), the ball Bq (a', r') exists and Bq(a',r') = {p^iar))-Bq{p{a'),r'))· • if Bq(a, r') exists for any 0 < r' < r, then Bq(a, r) exists and Bq(a,r)= (J Bq{a,r'). 0 <r'<r In particular, if we define the distance to the boundary (w.r.t. the norm q) dx,q ' • X — • (0, +oo], dx,q{a) := sup{r G (0, +oo]: Bq(a, r) exists}, a e X, 5) Notice that Bq (z0, +oo) := C".
- 20. 6 1 Riemann domains then for any a e X the maximal ball (w.r.t. the norm q) Bq{a) = Bx,q(a) := Bq(a, dx,q(a)) is well defined. Observe that if Y is a subdomain of X, then d(ytP|K)<q < dx<q on Y. Put Pa,q := (a)· The definition of dxiq(a) extends in the standard way to subsets of X, namely dx,q(A) := inf{Jx,q (x): χ e Λ}, A c X. If q(A) > 0,thenfor0 < r < dx<q(A), we can define the r-th hull (w.r.t. the norm q) of A, ' A M ) : = [ J Bq(a,r). aeA Finally, put Xoo := {a € X: dx,q(a) = +00}. Definition 1.1.7. Let us fix the following two important conventions related to the above notations. If q = II II is the Euclidean norm, then we skip (if possible) the index q and we substitute the letter Β by Β and d by p. Consequently, we will write Β(a, r), Βχ(α, r), B(a), Βχ(α), px(a), pX(A), etc. If q(z) = |z| = max{|zj I, · • · > |z«|} is the maximum norm, then we skip the index q and we substitute Β by P, e.g. we will write Ρ (α, r), Ρχ (α, r), Ρ(α), Ρχ(α), dx(a), dx(A),A(r pa,etc. Remark 1.1.8. (a) If (Χ, ρ) = (G, idc), where G is a domain in Cw , then dc,q coincides with the standard distance function to C" G in the sense of the norm q. Observe that in general we have: d(X,p), q < (rf(p(X),id),q) 0 P- Moreover, if X is univalent, then d(X,p),q = (rf(p(X),id),q) Ο p. (b) By Remark 1.1.2(f), if Xoo Φ then ρ maps homeomorphically X onto C" (in particular, X^ = X). In other words, except the trivial case where ρ maps bijectively X onto C", the set Xoo is empty. (c) If Xoo = 0, then dx,q(x) - dx,q(a) < q(/?0O - />(«)), α (Ε Χ, χ e Bq(a).
- 21. 1.1 Riemann domains over C 7 In particular, the function άχ,q is continuous. Λ. A. A Indeed, by Corollary 1.1.6(b), for any χ g Z?q (α) the set U := ßq (a) U Bq(x) is a univalent subdomain of X. Consequently, we have: dx,q(x) - dx,q(a) = ^([/.plyJ.qU) - ^(t7,p|c/),q(«)l = M(P(t/),id),q(pU)) -d(p(t/),id),q(/?(a))l < q(/>(*) - p(a)). (d) If (Xj, pj) is a Riemann domain over Cn >, qj : — • R+ is a C-norm, j = 1,2, andq(zi,z2 ) := max{qi(zi), q2(z2)h then d(x1xx2,pixp2),q(xi, X2) = min{d(xliPl),qi (*i), </(χ2 (*2)}, {x,xi) G Xi X x2. (e) Let Κ c X be compact. Then: • dx,q(K)> 0. • the set /sT(r,<,) is compact for any 0 < r < dx<q(K). Indeed, let ι C *υ g Bq(av,r), av g Κ, ν > 16) . Since /Γ is compact, we may assume that av —> ao e Κ and p(xv) —> zo <ξ C". Moreover, we may assume thatav G Bq(ao, ε), ν > 1, where ε := dxq{K) — r. Then Bq(av, r) C Bq(ao) and, consequently, = p^]q(p(xv)) — • P^]q(zo) =• *o- Moreover, q(p(*o) - p(ao)) < r. Hence xo G Bq(a0, r) C • dx,q(K^) > dx,q(K) - r (cf. Corollary 1.1.10). (f) If Κ is compact and univalent, then K(r 'q * is univalent for small r > 0. Indeed, suppose thatxv g Bq(av, 1/v), yv G Bq(bv, 1/v), av, bv g K,xv ^ yv, p(xv) = p(yv), ν » 1. We may assume that av —> ao e K, bv — • bo G K. Then p(ao) = ρ (bo) and therefore ao = bo. Thus xv, G Bq(ao), ν 1; contradiction. Proposition 1.1.9. IfX oo = 0, thenfor any a G X there exists a vector w G Cn with q(w) = dx,q(a) such that lim dx,q((pa,q)~l (p(a) + tw)) = 0. [U, 1 • 1 In particular, dx<q(Bq(a)) = 0. Proof. Fix an a G X. To simplify notation put Λ A d:=dx<q, Uo:=Bq(a), i/o := Bq(p(a),d(a)) = p(Ü0), s := (ρα,μΓι = (ρόοΓ1 . 6) One can prove (cf. Remark 1.1.14) that X is metrizable, and therefore, the compactness of JS-(r, q)may be checked via sequences.
- 22. 8 1 Riemann domains Suppose that for any vector w with <(w) = d(a) there exist e(w) > 0 and a sequence [0,1) 9 tk{w) / 1 such that d(s(p(a) + tk(w)w)) > e(w), k> 1. Now, for any w with q(u;) = d(a),fixafco = ko(w) such that (l—tko(w))d(a) < e(w) and let z(W) := p(a) + tk0(w)w e UQ, X(W) := s(z(W)) e UQ (note thatJ(jr(ii;)) > e(w)). In particular, 3i/o C ( J Bq(z(w),e(w)). q(w)=d(a) Since 3i/o is compact, there exist vectors wi , . . . , wμ with q(wi) = •• • = q (wn) = d(a) such that Ν dU0 C ( J Bq(z(wj),s(wj)). j=ι Put Üj := Bq(x(wj),s(wj)), Uj := p(Uj) = Bq(z(wj),s(wj)), j = l,...,N, and define Ν Ü := t/o U Uj. 7=1 Λ Λ — It is clear that U is an open neighborhood of a and p(U) D UQ. We will prove that U is univalent. Then, the ball Bq(a, r) would exist for some r > d{a), which contradicts the definition of d(a). First, observe that, by Corollary 1.1.6(b), for any j e {1,..., TV} the set Üoj := Uo U Üj is univalent. Moreover, for any j, k e {1,..., N}, if Üj Π Ük φ 0, then the set üj,k •- üj u ük is also univalent. Thus, it remains to exclude the situation where we have two points yj 6 ÜjÜ0,yk e ÜkÜ0 with/?()>;) = p(yk) and ÜjCÜk = 0. Since Uq Π Uj φ 0, UoHUk φ 0, and Uj DUk φ 0, there exists a point zo e UQ Π Uj Π Uk. Thens(zo) = ^ l f / 0 ; r l ( z o ) = 6 Üj a n d ' similarly, s(zo) e Ukcontradiction. •
- 23. 1.1 Riemann domains over C" 9 C o r o l l a r y 1 . 1 . 1 0 . Let Κ c X be compact. Then dx,q{K^) = dx,q(K) - r, 0 < r < dx,q(K) (cf. Remark 1.1.8(e)). Proof. We may assume that Xoo — 0· Fix Κ and r. We already know that dx,q(K(r 'q ^) > dx<q(K) — r. Since the function άχΛ is continuous, there ex- ists an a e Κ such that dx,q(a) — άχ^(Κ) =: R. By Proposition 1.1.9 there exists a w e Cn with q(u;) = R such that lim[o,i)3f^i dx,q(x(t)) = 0, where x(t) := (pa,q)~l (p(a) + tw). Let θ e (0, 1) be such that r = 6R. Put b := x(ß). Then b e K(r 'q) and x(t) e Bx<q(b, R - r) for θ < t < 1. Consequently, dx,q(b) = R-rmd hence dx<q(K^^) = dx,q{K) - r . • Remark 1 . 1 . 1 1 . Let γ: [ 0 , 1 ] —> p(X) be a curve. Fix an a e ρ~1 (γ(0)) and let Jo denote the set of all f e [0, 1] such that there exists a curve yt: [0, /] —> X with yf(0) = a and ρ ο yt = y |[o,r]- Then we have: • 0 e /o. • If t e /o, then [0, t] c Iq and therefore Iq is an interval. • yf/ = YT»[0J>] for 0 <t'< t", t" e IQ (by the identity principle for liftings) and, therefore, we have a lifting γ: IQ —> X of γ |/0 with p(0) = a. • IQ is open in [0, 1]. Indeed, if U is an open univalent neighborhood of Y(t) (for some ί e /ο Π [0, 1)) and 0 < δ < 1 - ί is such that γ ([ί, t + δ]) C p(U), then we can define: Μ ) , t g [0, t] (.pu)-l (y(t)), te[t,t + S]. Consequently, either IQ = [0, 1] (and then γ is a lifting of y ) or IQ = [0, t*) for some 0 < t* < 1. • If IQ = [0, /*), then dx(y(IO)) = 0 (in particular, y(IO) is not relatively com- pact). Indeed, suppose that άχ(γ(Ιο)) = r > 0. Take a point 0 < t < t* such that y(t) € P(y(t), r) for t € [t, i*] and define γ tit), t e [0, t] Yt+s(0 := { ,„,.Λ _ι( _ I Y i ( t l Pöi Then yt* is well defined and it is a lifting of / |[o,f*]; contradiction. Analogously as we defined the distance to the boundary (w.r.t. a norm), we can introduce the distance to the boundary in a given direction. For ζ ε Cn , ξ e Cn , and r > 0 let A^z,r) :=ζ + (ΓΕ)ξ (note that Δ0(ζ, r) := {z}).
- 24. 10 1 Riemann domains For a point a e X let Αξ(α, r) denote a univalent set containing a (if it exists) A such that ρ(Αξ(α, r)) = Αξ(ρ(α), r). A A Observe that the disc Αξ (a, r) exists for small r > 0. Moreover, if Αξ (a, r) exists, then it is uniquely determined (note that Δο(α, r) = {a}). Define the distance to the boundary in direction ξ, δχ^ : Χ —• (0, +oo], A δχ,ξ(α) := sup{r > 0: Αξ(α, r) exists}, a € X. A A A A Remark 1.1.12. (a) If Αξ(α, r) and Αη(α, s) exist, then the set Αξ(α, r) U Αη(α, s) is univalent (use Corollary 1.1.6(b)). (b) If Αξ(α, r) exists (ξ φ 0), then for any a! € Αξ(α, r) and 0 < r' < r — || p(a') - p(a)||/||£||, the disc Ας (a', r') exists and Α ξ ( α ' , / ) = ( ρ λ ξ Μ ) - ί ( Α ξ ( ρ ( α ' ) , / ) ) . If Αξ(α, r') exists for any 0 < r' < r, then Αξ(α, r) exists and Δ ξ(α,Γ)= ( J Α ξ ( α , / ) . 0 <r'<r A A (c) Assume that Δξ0(αο, R) exists. Fix 0 < ro < R. Put Κ := Αξ0(αο, ro) and let ε > 0 be so small that the set Λ := is univalent (cf. Remark 1.1.8(f)). Let U c X x C" χ R>o be an open neighborhood of the point (ao, ξο, ro) such that A$(p(a),r) C p(A) for any (a,%,r) 6 U. Then for any (a,£,r) e U the disc Αξ(α, r) exists and Δί (α,Γ) = (ρ|Α )-1 (Δξ (ρ(α),Γ)). In particular, the function X xCn 3 ( χ , ξ ) ^ δχ,ξ(χ) e (0, + o o ] is lower semicontinuous. (d) Let q: Cn —> R + be a C-norm. Then Bq(a, r) exists iff Αξ(α, r) exists for any ξ with q(£) = 1. Moreover, Bq(a,r)= (J Αξ(α,Γ). q«)=l Hence d x , q ( x ) = Μ{δχ,ξ(χ): ξ e C", q(!;) = 1}, χ 6 X. Notice that Proposition 1.1.9 says that for any a e X there exists a ξ € C" with q(£) = 1 such that dxiq(a) = δΧιξ(α).
- 25. 1.1 Riemann domains over Cn 11 Lemma 1.1.13. Let ξ e Cn , ||£|| = 1, and let q^(z):=max{±||z-<z,£)£||, (ζ,ξ)1 ze C (e > 0), where { , ) denotes the standard Hermitian scalar product in C" Then dx ,qfi£ / * δχ,ξ when ε 0. Proof. First observe that if L: Cn —> Cw is a C-linear isomorphism, then d(X,p),q = d(X L o p ) i t l o L -l, δ(Χ,ρ),ξ = hx,Lop)MH)· If, moreover, L is unitary and L(£) = en = (0,..., 0, 1), then q^e ο L-1 = qen,e· Thus we may assume that ξ = en. Obviously, dXi({en e < Sx,£n and dx, „ ^ , > dXt q ^£ „ when 0 < ε' < ε". Now fix an a e X, take 0 < r < δχ,βη(α), and let Κ := Αβη(α, r). Since Κ is compact and univalent, there exists 0 < η < dx(K) such that the set A := is also univalent. It is clear that there exists an ε > 0 such that U := Bqen ε(ρ(α), r) c ρ (A). Hence (pA)~l (U) = Bx^en e(a, r) and, consequently, dx^en C(a) > r. • For any curve γ: [0, 1] — • C" let L(y) denote the length of γ (w.r.t. the Euclidean norm), i.e. Ν L(y) := sup { Σ IIY(tj) - Y(tj-1)||: Ν e N, 0 = t0 < • • • < tN = l}. j=ι Now we introduce a structure of a metric space on a given Riemann domain (X, p). First, since X is arcwise connected, we conclude that for any x', x" e X there exists a curve γ: [0, 1] —> X with y(0) = x', y(l) = x" such that L(p ο γ) < +oo. Indeed, if γ: [0,1] —> X is an arbitrary curve with p(0) = x', p(l) = x", then we A find points x,..., xn-1 £ γ([0, 1]) such that xj- € M(xj), j = 1,..., N, where XQ := x', XN '•— X "• The curve defined as the union of the 'segments' is well defined and L(p ο γ) = Σ]=ι p(xj~i) — p(xj) < +oo. Now, we define the inner (arcwise) Euclidean distance on (X, p) σχ: Χ χ X —• R+ by the formula σχ{χ', χ") := inf{L(p ο γ): γ: [0, 1] — • Χ is a curve with p(0) = x', y (1) = χ"), χ', χ" € X. 7) (z,w) :=Ej=iZjWj. ^ Observe that ζ — (ζ, ξ)ξ is the orthogonal projection of the vector ζ on the complex hyperplane (Q)-1 -. Note that is a C-norm.
- 26. 12 1 Riemann domains Remark 1.1.14. (a) σ^ρ)(χ', χ") > σ^χ)^){ρ{χ'), Ρθ"))> x" e X. (b) If A C X is a univalent set such that ρ (Λ) is convex, then σχ{χ',χ") = ||pix') - p(x"), χ', χ" € Α. Λ (c) Βχ(α, r) = {λ € Χ : σχ(α, χ) < r}, a e Χ, 0 < r < ρχ(α). Indeed, the inclusion ' c ' follows from (b). Suppose that σχ(α,χ) < r. Then there exists a curve γ : [0, 1] —• X with γ(0) = α, γ(1) = χ, and L(p ο γ) < r. In particular, (ρ ο y)([0,1]) C M(p(a), r). Hence, by the identity principle for liftings, Υ '• [0, 1] —• Βχ(α, r). (d) σχ is a continuous distance. Observe that the topology generated by σχ coin- cides with the initial topology of X. (e) For any metric space (7, d) and mapping f : X —> Υ the following conditions are equivalent: (i) d{f(x'),f(x"))<ax{x',x"), x',x"eX (ii) d(f(x'),f(x))<p(x')-p(x), χεΧ,χ'eMx(x). In particular, by Remark 1.1.8(c), if Χ<χ, = 0, then Ipx(x') - pxix")I < σχ(χ', χ"), χ', χ" € Χ. The implication (i) (ii) follows from (b). Suppose that / satisfies (ii). Fix x', x" G X and a curve γ: [0, 1] —• X with γ(0) = λ', y(l) = χ". It is clear that there exist Ν e Ν and 0 = fo < · · · < = 1 such that γ(ί/) e Βχ(/(ί,_ι)), j = 1,..., N. Hence Ν d(f(x'), fix")) < ^d(f(Y(tj-1)), f(Y(tj))) 7=1 Ν < Ε WpiyVj-M ~ p(y(tj» < l (P ° y). j=ι which implies (i). (f) Let ao ε X, 0 < ε < ρχ (üq) and let Ue be the connected component of the set {x e Χ: ρχ (χ) > ε} with üq g Ue. Let συε be the inner Euclidean distance on (t/e, pue)· Then for any t > 0 the set Ue,t := {x eUg·. aUe(ao,x) < t} is relatively compact in X. Indeed, let I := {t > 0: U£J m X}. Observe that, by (c), (0, ε] c /. To prove that I = R>o it suffices to check that if f e I, t > ε, then t + ε/2 e I. Thus, it suffices to show that Ue,t+e/2C U Βχφ,ε). beU£,t
- 27. 1.1 Riemann domains over Cn 13 Take a point a e Uej+e/2 Uej and let γ: [0,1] —• Us be a curve with p(0) = ao, γ(1) — a, and L(p ο γ) < t + ε/2. Take a r € (0, 1) such that the point b := γ (τ) lies in Βχ(α, ε) ε/2). Then ί + ε/2> L(p ογ) = L(p ο γ|[0,τ]) + L{p ο p|[r,i]) > L(p ο y |[o,r]) + IIΡΦ) - Ρ(α)II > L(p ° Pl[0,r]) + ε/2. Consequently, L(p ο γ|[ο,τ]) < t. Hence b e U£it and a e Βχ(£, ε). Remark 1.1.15. (a) It is known (cf. [Die 1944], [Bou 1965]) that for any connected manifold Μ the following conditions are equivalent: (i) Μ is countable at infinity i.e. Μ = (J/^i where Kj is compact, j — 1 2 9 >· 1, i.,... , (ii) The topology of Μ has a countable base. (b) If Μ is a connected Riemann surface (i.e. a connected one-dimensional complex manifold), then Μ is countable at infinity; cf. [Rad 1925], see also § 1.11. Theorem 1.1.17 will show that the same property is shared by all Riemann domains over C". Exercise 1.1.16. Complete the following construction showing that there exists a simply connected two-dimensional complex manifold Μ which is not countable at infinity (cf. [Cal-Ros 1954]). Let A c C, A uncountable, Μ := A χ C2 , top Μ := {U CM: VfeA: {(x,y) e C2 : (t,x,y) € U} etopC2 }. Fori, t g A, s φ t, put <DJjf: C χ C* —> C χ C*, Φ5>ί(χ, y) := +*,?); Φίι? is biholomorphic and Φ"/ = ΦΜ . For (s, xs, yj), (t, xt, yt) G Μ define (s,xs, ys) ~ (t,xt, yt) ys) = (t,xt, yt)) or (s φί, ys = yt, s+xsys = t + xty?) ((j, ys) = {t, xt, yt)j or ( j φ t, ys, yt e C*, (xs, ys) = Φs,t(xt, ~ is an equivalence relation. Put Μ :— Μ/ ~ and let top Μ be the standard quotient topology, i.e. the strongest topology on Μ such that the canonical projection Μ 3 (t,x,y) [(*,*,?)] e Μ ^ Observe that, since Μ is locally compact, the following conditions are equivalent: • Μ is countable at infinity; • Μ = U/^i Uj, where Uj is relatively compact and open, j = 1,2,...; • Μ — U/ÜLi where Kj is compact and Kj C int Kj+, j — 1,2,
- 28. 14 1 Riemann domains is continuous; U e top Μ <(=?• VfeA: {(*,?) € C2 : [(?,*,>>)] € U) € topC2 . For t e A define Ut := Tim x C2 ), φ,: C2 —> t/„ <p,(x, y) := [(ί, λ, y)]. Then: • (pt is continuous and bijective. • U e top Μ Viey4: (£/ Π I/,) 6 top C2 . • Ut € top M. • is continuous. . M = U ( e A [ / ( , t o p M 6 r 2 . • (i/r, is an atlas of a complex structure on M. • Μ is connected. • The set S := π {A χ {(0,0)}) is discrete and #5 = #A. In particular, Μ cannot be countable at infinity. • Μ is simply connected. Theorem 1.1.17 (Poincare-Volterra theorem). Let (X, p) be α Riemann domain over Cn . Then X is countable at infinity. In particular, • for each ζ € p(X) the stalk p~l (z) is at most countable; • the topology ofX admits a countable base consisting of connected univalent and relatively compact sets. Proof. Fix a point oq G X and let Us and UEt, be as in Remark 1.1.14(f). Let ko € Ν be such that 1/ko < ρχ(αο). Put Vit := Ui/k,k, k > k0. By Remark 1.1.14(f) V* ^ X, k > ko. Obviously V^ c V^+i. It remains to observe thatx = u r = * 0 V 0 ) . Since X is countable at infinity, it is clear that each stalk ρ 1 (ζ) must be at most countable. Now we will directly show that the topology of X admits a countable base consisting of connected univalent and relatively compact sets. Let X = U ~ i where Kj is compact, j e N. For any j and for any rational number 0 < r < dx(Kj), consider the covering (P(x, r))x€Kj of Kj. Since Kj is A compact, we can select a subcovering (P(x, r))x€sjr, where c Kj is a finite set. We claim that the family 05 := (P(x, r));eNi ο<r<dx{K}), reQ, xeSj,r Take an arbitrary point a E X and a curve Γ: [0, 1] —> X with p(0) = OQ, y(l) = A, i := L{p ο γ) < +oo. Let 0 < ε < ρχ(γ([0, 1])). Then a e ί/ε>ί with t > I. Consequently, a e V^ with k > max{/co, 1/ε, I}. 1 ^ Independently of Remark 1.1.15(a).
- 29. 1.1 Riemann domains over C" 15 is the base we are looking for. We only need to prove that 05 is a base of the topology of X. Take an arbitrary open set U C X and a point a e U. Let 0 < R < άχ(α) be such that P(a, R) C U. Let jo be such that a e Kj0 and let 0 < ro < dx(Kj0), ro € Q, be such that ro < R/2. There exists an xo € Sy0>/-0 with a e P(;co, ro). It A A remains to observe that P(xo> r o) C P(a, R) C U. • The end of this section is devoted to continuity properties of families of liftings. Theorem 1.1.18 (Monodromy theorem). Let (X, p) be α Riemann domain over Cn . Let A be locally compact, connected, and locally connected topological space and let Β be a locally connected topological space. Fix an ao € A. Assume that F Αχ Β —• p(X) is a continuous mapping. Suppose thatfor each b e Β the mapping Ft, := F(·, b) has A A a lifting Fb: A —> X such that the mapping Β 3 b —> />(ao) e X is continuous. Then the mapping Α χ Β 3 (a, b) —Fb(a) e X is continuous. Proof. First observe that F is continuous at each point (ao, b), b e B. Indeed, fix a b e Β and let W be a univalent neighborhood of F(ao, b ). There exist connected neighborhoods U c A of ao and V c Β of b such that: F({ao] χ V) c W (here we use the continuity of the mapping b —> F(ao, b)), F(U xV)c p(W). Define σ := (ρψ)~1 °Fuxv • Then for arbitrary b € V the mappings U 3 a —• F(a,b) and U 3 a —> a(a,b) are liftings of the mapping U 3 a —> F(a, b) that coincide at a = ao- Consequently, by the identity principle for liftings, they coincide A A everywhere. This means that Fuxv = o. In particular, F is continuous at (ao, ^l)· Now, fix an arbitrary point (a, b) e Α χ Β. We will prove that there exist neighborhoods U C A of at V c Β of b, W C X of F(a, b) such that W is univalent, F(U χ V) c p(W) and Fuxv = (plw)- 1 ° FuxV (which obviously implies the continuity of F on U χ V). For any χ e A let Wx C X be an arbitrary univalent neighborhood of Fb^x). Choose a relatively compact neighborhood Üx c A of χ such that Ft>l (Üx) (s Wx. Since F is continuous at (ao, b), we may assume that F(Üao χ V) C Wao for a neighborhood V c Β of b. Since A is a connected and locally connected Hausdorff space, there exist points x,..., χχ e A and domains U,..., U^ such that jci = ao, ai 6 UN, Uj C ÜX], j = 1,..., N, and Uj nUj+i φ 0, j = 1,..., Ν - 1 (cf. [Bou 1965]). Put Wj := WXj, j = ,...,N. Since F(Uj χ {^i}) = p{Fbx{Vj)) m p(Wj), j = 1,..., N, there exists a connected neighborhood V C V of b such that F(Uj χ V) c p(Wj), j = 1 ,.,.,Ν. Note that F{UX χ V) C W. Define Oj = (pwj)~l ° Fujxv Uj χ V —> X. It suffices to show that for any j = l,...,N: σ ί = FUjxV- (*)
- 30. 16 1 Riemann domains Observe that σ) = F on Uj χ {b}, j = I,..., N. Obviously (*) holds for j = 1. Suppose that (*) is true for j = ... ,k for some 1 < k < Ν — 1. Fix a point ε Uk Π Uk+i. Note that F(xo, b) = a^ixo, b), b e V. Fix a b e V. Observe that the mappings Uk+1 3 λ: —> F(x, b), Uk+ι 3 x —> Ok+{x, b) are two liftings of the mapping [/*+1 3 x — • F(x,b). Thus, by the identity principle for liftings, it suffices to prove that a^ixo, b) = F(xo,b) = σ*+ι(χο- b). For, observe that the mappings V 3 y —a^ixo, y) and V 3 y —> σ£+ι(χοι >0 are two liftings of V 3 y —> F(xo, y) that coincide at y = b. Hence, once again by the identity principle for liftings, we get the required equality. • In the case where A = Β := [0, 1], ao := 0 we get the following Theorem 1.1.19 (Classical monodromy theorem). Let (Χ, ρ) be α Riemann domain over Cn, xo ^ X' and let F: [0, 1] χ [0, 1] —> p(X) be a continuous mapping such that F{ 0, u) — p{x o), u € [0, 1]. Suppose thatfor each u € [0, I] there exists a lifting Yu [0, 1] —> X of the curve F(·, u) such that y«(0) = xo. Define F(t, u) yu(t), Λ t,u 6 [0, 1]. Then F is continuous. In particular, if there exists a point z € p(X) such that F(l,u) = z for all λ -j u e [0, 1], then the mapping [0, 1] 3 u —>· F(l,u) € ρ (zi) is continuous. Consequently, since the stalk p~l(zi) is discrete, there exists a point X[ € p~^(z) such that F{, u) = x for all u e [0, 1]. Α Riemann domain (X, p) is said to be a cover if for any point zo € p(X) there exists an open neighborhood UQ such that each connected component U of the set p'HUo) is univalent and p(U) = UQ. Α Riemann domain (X, p) is said to be arbitrarily continuable if for any curve γ: [0,1] —> p(X) and for any a e ρ~ι(γ(0)) there exists a lifting γ: [0,1] —> X with y(0) = a. Notice that in fact (X, p) is arbitrarily continuable iff (X, p) is a cover; cf. the following exercise. Exercise 1.1.20. Let (X, p) be a Riemann domain over C". Prove that the following conditions are equivalent: (i) (X, p) is a cover; (ii) (X, p) is arbitrarily continuable; (iii) For any metric space Τ such that • Τ is arcwise connected, • Γ is locally arcwise connected, • Τ is homotopically simply connected, for any continuous mapping γ: Τ —>· p(X), for any ίο € Τ, and for any a e p~l(y(to)), there exists a lifting γ: [0, 1] —> X such that γ (to) = a Notice that in (ii) we have simply Τ : = [0, 1].
- 31. 1.1 Riemann domains over C" 17 (iv) For any C-norm q: Cn —> R+ we have d(x p ) ^ = (d(P(X),id),q) ο ρ (cf. Remark 1.1.8(a)). Hint, (i) =>• (ii): Take an arbitrary curve γ: [0,1] —> p(X) and a point a e ρ~ι (γ(0)). Let /o be as in Remark 1.1.11. Since (X, p) is a cover, the set /o is closed. Hence Iq = I. (ii) =Φ· (iii): For any t e Τ let at: [0, 1] — • Τ be an arbitrary curve with σ,(0) = ί0» σ,(1) = t- Put 8t := γ ο σ,: [0, 1] — • p(X) and let St: [0, 1] —> X denote the lifting of 8t with <5,(0) = a. The classical monodromy theorem implies that is independent of at . Define γ{ΐ) := <5,(1). Now it remains to prove that γ is continuous. (iii) (iv): Fix a C-norm q: C — • R+ and a e p_ 1 (zo)· Let Τ := Bq(p(a), r) c p{X), γ := idr . By (iii) there exists a section γ : Τ — • X of (Χ, ρ) over Τ such that γ (p(a)) = a. Then y(T) = Z?q(a,r). (iv) =>• (i): Fix a z 0 e p(X) and letl(z0 , 2R) c p(X). Put Uq := B(zo, r). Let U be a connected component of p~l (Uo). Then U = Μχ(α, r), where a e p~l (zo)· In particular, pu maps homeomorphically U onto Uq. Notice that the assumptions on Τ may be weakened. Proposition 1.1.21. Let (X, p) be a cover over C". Then: (a) #p~x {z) = #p~l {zi) for any z, zi e p(X). (b) If ρ(X) is simply connected, then ρ is injective (i.e. X is univalent). Proof, (a) Fix Z,Z2 £ p(X), z φ Z2, and let γ: [0, 1] —> p(X) be a curve such that y(0) = zi, y(l) = zi. It suffices to prove that #p~x (z) < #p- 1 (^2)· For any χ e p~l (z) let f(x) = γχ(1), where γχ is the lifting of γ with yx(0) = x. It is clear that / : p~l (zi) —> p~l (z2)· By the identity principle for liftings the mapping / is injective. (b) Suppose that αι,α2 € X are such that/?(a i) = p(a2) =: zoandleta: [0,1]—• X be a curve such that σ(0) = αϊ, σ(1) = α2· Let F : [0, 1] x [0, 1] —> p(X) be a continuous mapping (a homotopy) such that • F(0, .) = F(l,-) = zo, • F(·, 0) = ροσ, • F(; 1) = zo· Since (X, p) is arbitrarily continuable, for each u e [0, 1] there exists a curve F(·, u): [0,1] —> X such that F{0, u) — a. In particular, by the identity prin- ciple for liftings, F(·, 0) = σ, F(·, 1) = ai and ρ ο F(-, ü) — F{-, u). By the monodromy theorem, a2 = F( 1,0) = F(l, 1) = a. •
- 32. 18 1 Riemann domains 1.2 Holomorphic functions The aim of this section is to collect some basic properties of holomorphic mappings between Riemann domains. The local theory of holomorphic functions on Riemann domains over C" is of course the same as on domains in C". Thus the stress will be put on global properties. Definition 1.2.1. Let (Χ, ρ) be a Riemann domain over Cn . A function / : X —• C is said to be holomorphic ( / e Θ(Χ)) if for each open univalent subset U C X the function / ο (/>|ι/)-1 is holomorphic in the standard sense on p(U) c C" (equiv- alently: / is holomorphic in the sense of the complex manifold structure of X; cf. Remark 1.1.2(c)). A mapping F = (F,..., Fm): X —> Cm is holomorphic (F e 0(X, Cm )) if the functions F,..., Fm are holomorphic. If (7, q) is another Riemann domain over Cm , then acontinuous mapping F: X —> Y is said to be holomorphic (F e Θ(Χ, Y ) ) i f q o F e Θ(Χ, Cm ) (equivalently, F is holomorphic in the sense of the complex manifold structures on X and Y)13) . The space Θ(Χ) will be always endowed with the topology τχ of locally uniform convergence, i.e. the Frechet space topology generated by the seminorms 0 ( X ) 3 / — • II/|| a:,·, ./ = 1 , 2 , . . . , where λ is any sequence of compact subsets of X such that Kj c int Kj+1 and X = U£li Kj (cf. the Poincare-Volterra theorem). Exercise 1.2.2. Prove that τχ is a Frechet space topology. Prove, moreover, that τχ is independent of the particular choice of a sequence of compact sets. Observe that an analogous topology may be introduced in the case where X is a connected complex manifold, countable at infinity. For / e Θ(Χ) define the j-th partial complex derivative at a: df 9 ( / o (pit/)"1 ) — (a):= (p(a)), j = l , . . . , n , azj oZj where U is an arbitrary open univalent neighborhood of α 6 X Observe that the definition of ^ ( a ) is obviously independent of U. 13 ' Observe that if Y is non-univalent, then there exists a discontinuous mapping F: X —>· Y such that q ο F = const. In the above formula, the operator ^ on the right hand side denotes, of course, the standard y'-th partial complex derivative in Cn .
- 33. 1.2 Holomorphic functions 19 For ν = (vj,..., vn) e Z" define the v-th partial derivative of / at a: / 9 Vl / d v " 0 7 ) « , ) = ΟΪ/)(α) := ( - ) o . . . o ( - ) /«,). Clearly 3υ / e Θ(Χ). By the Weierstrass theorem, the operator 0(X) 3 f ^ d v f e 0(X) is continuous in the topology τχ. We have the following Cauchy inequalities: 11971k < K<£X,0<r< dx(K), f € <9(X), ν e 7L. The power series Tafiz) := Tpia){f ο p-l )(z) = Σ - P^y veZn + is called the Taylor series of / at a. Let d(Taf) be the radius of convergence of the Taylor series T a f , d{Taf) := sup{r > 0: Taf(z) is convergent for ζ e P(p(a), r)}. Observe that d{Taf) > dx(a) and fix) = Taf{p{x)), f e Θ(Χ), aeX, x€ Ρ (α). Proposition 1.2.3 (Identity principle). Let (X, p), (F, fee Riemann domains over C" andCm , respectively. LetF, G € Θ{Χ, Y) and assume that F = G on anon-empty open subset. Then F = G on X l5 Proof. Put Xo := {JC e X : F = G in an open neighborhood of JC}. Clearly, Xo is non-empty and open. It suffices to show that XQ is closed. Let a € Xo· By continuity we get F(a) — G(a) —: b. Let U and V be univalent neighborhoods of a and b, respectively, such that F(U) C V and G(U) c V. We may assume that U is connected. Then the holomorphic mappings / := (qv) ο F ο (ρν)- g := (qv) ο G ο (plu)'1 , defined on p(U), coincide on p(Xο Π U) Φ 0. Hence, by the standard identity principle, they coincide on the whole p(U). In particular, F = G on U. Thus U C XQ. • ' The result remains true in the case where X and Y are connected complex manifolds — Exercise.
- 34. 20 1 Riemann domains Definition 1.2.4. Let !F c O(X) be a vector subspace. Assume that Τ is endowed with a Frechet space topology τ. We say that F is a natural Frechet space if the convergence in the sense of τ implies the convergence in the sense of τχ, i.e. the identity embedding {Τ, τ) —(0(X), τχ) is continuous. Obviously, (Θ(Χ), τχ) is a natural Frechet space. The following example shows that many classical spaces of holomorphic functions are natural Frechet spaces. Example 1.2.5 (Natural Frechet spaces), (a) Jf°°(X) := the space of all bounded holomorphic functions on X with the topology generated by the supremum norm || ||χ. Notice that (M°°{X), || ||χ) is even a Banach algebra. (b) Lj(X) := Θ(Χ) Π LP (Χ) = the space of all p-integrable holomorphic func- tions with the topology induced by the norm || lp(X) (1 < ρ < oo). Here LP (X) is taken w.r.t. the Lebesgue measure A — Αχ on X constructed as follows: A set A C X is called measurable (A e £(X)) if p(A Π U) is Lebesgue measurable in C" for any open univalent set U C X. It is clear that any Borel subset of X is measurable. One can easily check that a set A c X is measurable iff any point a e X has an open univalent neighborhood U such that p(A Π U) is measurable. Let X = xUj, where each U/ is open and univalent (use the Volterra theorem). Put Bi := Uu Bj := Uj {U U · • · U Uj-i), j =2,3,.... For A e £(X) we put A*(A) := ΣΤ= ι A2n(p(A Π Bj)). One can prove (Exercise) that Αχ is a regular Borel measure on £(X) which is independent of the choice of a sequence (i//)?^. Moreover, Λχ(Α Π U) = Λ2η (ρ(Α Π U)) for any measurable set A c X and open univalent set U c X. Observe that if A e £(X) and / : A —> [0, +oo] is a measurable function, then fA fdAx = J^JLi ΙΡ(απβ ) f 0 (Puj)~l dA2 n, where Uj and Bj, j = 1, 2,..., are as above. By the Cauchy integral formula, we get UWK < — W f fdA, f G O(X), K m x , 0 < R < dx(K). (πr1 )" JK(r) Hence, UWK < 2 B ll/ll£P(X), / e Lp h(X), K m X , 0 < R < dx(K), where l/p + 1/q = 1. In particular, L%(X) is closed in LP (X), which shows that {Lp h{X), II ||lp(X)) is a natural Banach space. Obviously, L%>(X) = 3t°°(X). Note that L2 h (X) with the scalar product (/, «)—•/* fidA Jx is a Hilbert space.
- 35. 1.3 Examples of Riemann regions 21 (c) Λ(Ω) := Θ (Ω) Π <3(Ω), where Ω is an open subset of X, with the topology generated by the seminorms / —> /κ with Κ being a compact subset of Ω. Observe that if Ω m X, then Λ(Ω) is a subalgebra of Μ°°(Ω). (d) &(k) (X, δ) := {/ e Θ(Χ): Sk fx < +00} = the space of all δ-tempered holomorphic functions on X of order (degree) < k with the norm / —> ||5*/||χ, where δ: X —> (0, +oo) is such that /δ is locally bounded (k > 0). Note that Oik) (X, δ) is a Banach space and that 0(O) (X, δ) = J{°°(X). Exercise 1.2.6. (a) Let (!Fi, r,-)ie/ be a countable family of natural Frechet spaces in Θ(Χ). Put Τ = Π,6 / Ti and let τ be the weakest topology on Τ such that all the mappings id: Τ —> Ti,i e I, are continuous (i.e. fk —^ fo in τ : <£=>- fk —> /o in τ, for any i e I). Prove that (Τ, τ) is a natural Frechet space. (b) Let Σ c Z " , 0 e Σ, and let (!FV, τν)ν€γ,, be a family of natural Frechet spaces. Define f | / e f'0 : v veE : 3v f e -fy}. We endow Τ with the weakest topology τ such that the operators 3V : Τ —>- ν e Σ, are continuous (i.e. fk —>• fo in τ : <==>• 3v fk —> 3v /o in τ ν for any ν e Σ). Prove that (!F, r) is a natural Frechet space. (c) Observe that using (a) and (b) we can produce a lot of new natural Frechet spaces. For instance, with Σ := {υ e Z" : |v| < k}, we get: Jf°°'k (X) := {/ e J£°°(X) : 3 v f e M°°(X), |v| < it}, Lp h'k (X) •= if e Lp h(X): 3γ e Lp h{X), |v| < *}, Ak (Q) := {f e Θ(Ω): 3v f e *Α(Ω), |v| < k}, k e Z+ U {00}. Obviously, J£°°'0 (X) = Jf°°(X), LP '°(X) = Lp h{X), and Λ°(Ω) = Λ(Ω). 1.3 Examples of Riemann regions Example 1.3.1 (The sheaf of germs of holomorphic functions; see Example 1.6.6 for a more developed situation). Let a e Cn . Define Ga :={(U,f): C/€«(fl), / e 0(U)}, where 55(a) denotes the family of all open neighborhoods of a. For (U, /), (V, g) e Θα we put (U, f ) ~ (V, g) 3Wef8(a): W C U Γ) V, fw = gw- It is clear t h a t i s an equivalence relation. Put Θα :=Oa/~.
- 36. 22 1 Riemann domains A The class [(i/, f)]a is called the germ of / at a. We write fa := [(£/, /)]a. Define KU, f ) h + [(V, g)h ••= l(u nv,f + g)h, [(£/, f ) h · [(V, g)h ·.= [(£/ η v, / · g)h. One can easily check that the operations +, : Θα χ Θα —> Θα are well defined and that (Θa , +, ·) is a commutative ring with the unit element (the ring of germs of holomorphic functions at a). Let f = fa e Θa. Then f(a) := f(a) and dv f := [(U, 3v /)]„ are well defined. In particular, we can define Taf(z) := T ^dv f(a)(z-a)v = Taf(z). —' ν! The mapping Θα β f —> Τa f € the ring of all power series with center at a that are convergent in a neighborhood of a is an isomorphism. Put Θ = (J Θα χ {a} aeC and let π: Θ —> C" be given by the formula 7r((f, a)) = a (in the sequel we will denote elements of Θ either formally as pairs (f, a) (when we want to point out that f € Θa), or simply as germs f). For f € Θα and for any (U, / ) € f define Uf(U, f )•.={ ( f z , z ) : z e U ) c 0 . It is clear that the system (ilf (U, / ) ) a € c n , feöa , (t/,/)ef is a neighborhood basis 1 6 We endow Θ with the topology generated by this basis. We will show that this is a Hausdorff topology. Take (f, a), (g, b) e Θ, (f, a) φ (g, b). If α φ b, thenllf (U, f ) Π Ü0(V, g) = 0 provided that U Π V = 0. If a = b, then Uf(U, f ) Π il9 (U, g) = 0 if U is an open connected neighborhood of a (by virtue of the identity principle, if fZQ = gzo for some zo € U, then f = g and therefore f = g). Let X be a set. Suppose that for each χ e X we are given a non-empty family B(x) of subsets of X such that • χ e U for any U e B(x), • for any y e U e B(x) there exists a V e B(y) such that V C U, • for any U, V e B(x) there exists a W e B(x) such that W C U η V. The above system (B(x))x e x induces a topology Τ on X: we say that a set Ω c X is open (Ω e T) if for any χ e Ω there exists a U € B(x) such that U C Ω. Notice that B(x) c Τ, χ e X. Thus the system (B(x))X G x is a basis of open neighborhoods in T.
- 37. 1.3 Examples of Riemann regions 23 Observe t h a ^ is continuous and π is injective on any neighborhood 11 = ilf (U, / ) . Moreover, π|n: U —• U is homeomorphic and Or|u)_1 (z) = (Λ, ζ), ζ e U. Thus (Θ,π) is a Riemann region over C". It is called the sheaf of germs of holomorphic functions in C". Notice that d0((f,a))=d(Taf), f e 0 a , a e Cn . Let γ: [0, 1] —• C" be a curve. Suppose that for each t e [0, 1] we are given a convergent power series σ (t) with the center at γ (t) (or, equivalently, a germ o(t ) from 0K(,)) such that the mapping [0,1] 3 t —> y(t) := (σ(ΐ), y(t)) € 0 is continuous. Then we say that the series σ(0) can be continued analytically along the curve γ (to the series σ()). In other words, the series σ(0) can be continued analytically along γ if there exists a lifting γ : [0, 1] —• Θ of γ such that y(0) = (σ(0), y(0)). Remark 1.3.2. (a)DefineF: 0 —• C,F((f,z)) := f(z). Then for any it = ilf (i/, / ) we get F ο (ττ |u)_1 = / on U. This shows that F is holomorphic on Θ (F e 0(0)). (b) For any open set Ω c C and for any / e 0(Ω) define s/: Ω —• Θ, A Sf(z) := ( f z , ζ), ζ € Ω. It is clear that sj is a section of (0, ττ) over Ω. Conversely, if s: Ω —• 0 is a section, then the function / := F ο s is holomorphic on Ω and s = sf. Thus, Γ(Ω, (0, ττ)) ~ 0(Ω). (c) Let G C C" be a domain, let / e 0(G), and let / denote the connected component of 0 that contains Sf(G). Suppose that for some a e G there exists a curve γ: [0, 1] —• C" such that }/(0) = a and that the series σ(0) := Taf can be continued along γ to a series σ(1). Since γ: [0, 1] —• 0 is a curve and p(0) g Sf(G), we conclude that y([0, 1]) c / , which shows that σ(1) e / . (d) Let dv : 0 —• 0, 9v (f, a) := (dv f, a). Then dv is continuous and π ο dv = π. In particular, 3V is locally biholomorphic (and, consequently, open). Note that 9υ (5/ (Ω))=59 ,/ (Ω). (e) Let η = 1 and let Λ C 0 be a domain. Define :={[(V,g)h: 3[(u,f)]atA: f : U ^ V is bijective , f(a) = b, g = f~1 }. One can easily prove that C 0 is also a domain (Exercise). In particular, if / : G —> D is biholomorphic (D, G C C), then (s/(G))_1 = sf-i(D). Example 1.3.3 (Riemann surface of /"1 ). Let G C C be a domain and let / : G —• D be a locally biholomorphic mapping, D := f(G). Evidently, (G, f ) is a Riemann domain over C. Moreover, ((s/(G))-1 , π) is a Riemann domain over C. Let X := {(u;, ζ) e G χ C: f(w) = z}. We endow X with the topology induced from C2 . Let p(w, z) :— z, (w, z) e X. Then, by the implicit function theorem, ρ: X — C is locally homeomorphic. Thus (X, p) is a Riemann domain over C.
- 38. 24 1 Riemann domains We say that Riemann domains (Χι, p), (X2, pi) are isomorphic if there exists a homeomorphic mapping a: X — • X2 such that p2 ο a — p; cf. § 1.4. We claim that the Riemann domains (G, / ) , (X, p), and {{Sf{G))~x , π) are iso- morphic. Let φ: G —» Χ, φ(νυ) := (w, f(w)). Then φ: G —» X is homeomorphic ζ) — w) and ρ ο φ = f . This shows that (G, / ) and (X, p) are isomorphic. Now let ψ: X — • (sf(G))~l , yjr{w, z) (gz, z), where g is a local inverse to / such that g(z) = w. It is clear that ψ is a homeomorphism (ψ~ι (§ζ , ζ) = (g(z), ζ)) and π οψ = p. Thus (Χ, ρ) and ((s/(G))_ 1 , π) are also isomorphic. Observe that the function s := φ~1 is holomorphic on X and / ο s = p. The domain (X, p) is called the Riemann surface of / - 1 . We would like to mention the following two particular cases. 1°: f ( w ) = wk ,w€G := C* (k g N). We already know that the domains (X, p) and (C*, / ) are isomorphic. Neverthe- less, since ρ coincides with the standard projection, the domain (X, p) has a more geometric nature. For any point zo € C* the stalk p~l (zo) consists of exactly k points, say Ao,..., Ak-1, Aj = (woj, zo), j = 0 , . . . , k - 1. Note that {wo,o, · · ·, m,k-i} = V^ö- Let U C C* be any simply connected open neighborhood of zo and let fo, ... ,fk-i e 0(U) be holomorphic branches of the k-th root such that fj(zo) = WQJ, j = 0 , . . . , k - 1. Then for each j = 0 , . . . , it - 1 the set Uj := {(/;·(ζ), ζ): ζ e U} is a univalent neighborhood of Aj and (ρυ})~1 (ζ) = ( f j ( z ) , z ) , ζ e U. In par- ticular, let zo := 1 > wo,j '•= exp(2nij/k), j = 0 , . . . ,k — 1, U := C Let y(t) = φ(οχρ(2πίΐ)), t e [0, 1] (recall that^(w;) = (w, wk )). Then γ: [0, 1] —> X is a curve such that Y(j/k) = Aj, j = 0 , . . . , k — 1, y(l) = y(0) = Ao- It is clear that y(t) 6 Uj for t e 7), where T0 := [0, U l] and Tj := j = l , . . . , k - l . This leads to the following geometric interpretation of the domain X. We take k copies of the domain C say Co,..., Cjt_i. Let ζ < £ Cj be such that Re ζ < 0 . We say that ζ lies on the 'upper part of Cj' if Im ζ > 0. Otherwise, we say that ζ lies on the 'lower part of Cj'. Now we glue the domains Co,..., Ck- together, crosswise along R_. That is, first we glue Co to C in such a way that we join the upper part of Co with the lower part on C1, next we glue the upper part of C to the lower of C2, and so on. Finally, we glue the upper part of Ck- to the lower part of Co- One can prove that the above construction is impossible in R3 without self inter- sections. 2°: f(w) := exp(u;), w e G := C. The geometric interpretation of (X, p) is following. We take a countable family (Cj)jez of copies of C Next, for each j e Z, we glue Cj to Cj+ crosswise along M_, joining the upper part of Cj with the lower part of Cj+1. Observe that this construction can be done in R3 .
- 39. 1.4 Holomorphic extension of Riemann domains 25 1.4 Holomorphic extension of Riemann domains To compare different Riemann domains over Cn we need the following notion of a morphism (generalized inclusion). Definition 1.4.1. Let (Χ, ρ), (Υ, q) be Riemann domains over Cn . A continuous mapping φ: X — • Y is said to be a morphism if q ο φ = p. If φ: (Χ, ρ) —> (Y, q) is a morphism such that φ is bijective and φ~λ : Y —> X is also a morphism, then we say that φ is an isomorphism. Obviously, the above notions extend to the case of Riemann regions. Observe that if G, G2 C Cn are domains and φ: (G1, idc,) —> (G2, idG2) is a morphism, then G1 C G2 and ψ = idG,g2 - Remark 1.4.2. Let (Χ, ρ), (Y, q) be Riemann domains over C" and let φ: (Χ, ρ) —> (Y, q) be a morphism. (a) By the identity principle for liftings, if ψ: (Χ, ρ) — • (Y, q) is another mor- phism with φ{α) = ψ(α) for some a € X, then φ = ψ. (b) If ψ: (F, q) —> (Z, r) is a morphism, then ψ ο φ: (Χ, ρ) —> (Ζ, r) is a morphism. If ψ: (Y, q) —> (Χ, ρ) is a morphism such that ψ ο φ(α) = a for some a € X, then φ is an isomorphism and ψ = (use (a)). (c) φ is locally biholomorphic. In particular, φ is an open mapping. (d) φ is an isomorphism iff φ is bijective (use (c)). (e) If Λ c X is univalent, then φ{Α) is univalent. In particular, φ(Ψχ(χ, r)) = Ψγ(φ(χ), r), χ G X , 0 < r < d x { x ) . Consequently, • dy ο φ > άχ, • if φ is an isomorphism, then dy ο ψ = άχ. (£)ϊ άγ ο φ = άχ, then φ(Χ) = Υ. Indeed, it is sufficient to show that φ(Χ) is closed in Y. Let _yo £ ψ(Χ)· Take 0 < 2r0 < dY(yo). Let y = φ(χ) € FVCyo. ro)· T h e n d x { x ) = dY(y) > r0 and, therefore, e P y ^ , r0) = φΦχ(χ, r0)) C <p(X). (g) The mapping φ*: Θ(Υ) Θ(Χ), <p*{g)-.= go<p, is an injective algebra homomorphism.
- 40. 26 1 Riemann domains Sometimes, to simplify notation, we will write /φ :=(<p*)-l (f), fe<p*(0(Y)). (h) φ* ο = dv x ο ψ*, ν e Ζ. In particular, Τφ(χ)/ = Tx(f ο φ), f e (9(F), χ eX. Indeed, using (e) we get pa = qV(a) ° φ and hence, for g e Θ(Υ) and a e X, we have: dv Yg(<p(a)) = dv (g ο q~(a))(q(<p(a)) = dv ((g ο φ) ο ρ~Χ )(ρ{α)) = dv x{g ο φ)(α). (i) In view of (h) we get d(Taf) > dY(<p(a)) for any a e X and / € φ*(Θ{Υ)). Now we are in a position to define the fundamental notion of holomorphic ex- tendibility in the category of Riemann domains over Cn . Definition 1.4.3. Let (X, p) be a Riemann domain over C" and let 0 Φ S C <9(X). We say that a morphism a : (X, p) —> (F, q) is an £-extension if 4 C a*(0(Y)) (i.e. for each / e S there exists age Θ(Υ) such that g ο or = /). If S = Θ(Χ), then we say that α: (Χ, ρ) —> (Y, q) is a holomorphic extension. Obviously, if α: (Χ, ρ) —• (Ύ, q) is an ^-extension, then it is an ^'-extension for any 0 Φ S' C S. Consequently, the most important is the case S = &(X). The above definition of the S-extension generalizes in a natural way to the case of Riemann regions over C" but in this case, to avoid situations where the region (F, q) is 'too large', we assume that every connected component of F intersects φ(Χ) (which is equivalent to the injectivity of the mapping φ*). We say that a family 0 φ S c 0(X) is d-stable if dv xf e S for any / e S and for any ν e Z" (equivalently, ^ e $ for any / e S and j = ... ,n). Remark 1.4.4. Observe that for any family S c Θ(Χ), the smallest 3-stable subal- gebra of Θ(Χ) with 4 U {pi,..., pn) c [4] may be described in the following way17) : [S]:={P(fi,...,fN): Ν eN, Ρ e J>(CN ), 18) where *'--={dv xf: f €<SU{Pu...,Pn}, veZn +}. 17 > Notice that 1 e [4]. denotes the space of all complex polynomials of Ν complex variables.
- 41. 1.4 Holomorphic extension of Riemann domains 27 Remark 1.4.5. Let a: (X, p) —> (Υ, q) be an -i-extension. Put 4a := ( a V t f ) = { f a : / € 4} = {g e Θ{Υ): g ο a e 4}. (a) If β: (Υ, g) — • (Z, r) is an ^"-extension, then β ο a : (Χ, ρ) — • (Ζ, r) is an -^-extension. (b) If β (Y, q) —> (Z, r) is a morphism such that β ο a: (Χ, ρ) —> (Ζ, r) is an -extension, then β: (Y, q) —> (Z, r) is an 4a -extension. (c) If 4 = (9(X), then 4a = Θ(Υ). (d) 4 is a vector space (resp. an algebra) iff 4a is a vector space (resp. an algebra). Moreover, by Remark 1.4.2(h), dv Yfa = { d v x f f . In particular, 4 is -stable iff 4a is 3y-stable. (e) d ( T a f ) = d(Ta(a)fa ) > dY(a(a)) f o r any α € X and / e 4. (f) a : (X, p) —> (Y,q) is an [^-extension (cf. Remark 1.4.4). Therefore, in the theory of holomorphic extension we can always assume that [4 — 4, i.e. 4 is a d-stable algebra with p , . . . , pn e 4. If 4 = Θ(Χ), then g(Y) = g(a(X)) for any g e G{Y). Indeed, suppose that a e g(Y) g(a(X)). Let j ·— g ο a — a Then ( g o a - a ) - f = 1 on X, and therefore, by the identity principle, ( g - a ) - f a = 1 on 7; contradiction. (h) If 4 = Je°°(X), then ||g||y = ||g oax, g e 4a . Consequently, 4a = M°°(Y) and a*: M°°{Y) —• M°°{X) is an isometry of Banach algebras. Indeed, suppose that there exists an α € g(Y) with a > ||g ο α||χ. Then the function / defined as in (g) belongs to M°°(X) and we conclude the proof as in (g). (i) Suppose that 4 is a natural Frechet space (cf. Definition 1.2.4). We already know (by (d)) that 4a is a vector subspace of Θ{Υ). Now we introduce a topology on 4a : Assume that the topology τ of 4 is given by a sequence of seminorms such that qj < qj+i. Let (Lj)j^j be an arbitrary sequence of compact subsets of Y such that Lj C i n t L ; + 1 a n d U ^ , ^ = Put ij := qj ο a*; observe that q; is a seminorm on 4a , j e N. Now consider on 4a the following family of seminorms I I L , ) ^ ! · This family defines a metrizable locally convex topology xa on 4a such that («ν go) «=>• (gv ° « go ο α and gv g0).
- 42. 28 1 Riemann domains In particular, the topology τα is independent of the choice of admissible sequences Wjii> (W)jLv Observe that τα is a topology of a Frechet space. Indeed, if (gy)^j is a Cauchy sequence in (Sa , τα ), then (gv ο is a Cauchy sequence in (S, r) and ( g v ) ^ is a Cauchy sequence in (0(F), τχ). Hence there exist functions /o € S and go £ Θ(Υ) Τ Τγ such that gvoa —> /ο and gv —> go· It remains to prove that goοα = fo• Recall that τ is stronger than τχ. Hence, for any χ e X, we get fo(x) = limy^+00(gv ο a)(x) = g0(a(x)). The mapping a*: (Sa , τα ) —> (S, τ) is obviously continuous (recall that it is an algebraic isomorphism). Since (£a , τα ) is a Frechet space, the Banach theorem implies that a*: (<8a , τα ) —>· (S, τ) is a topological isomorphism, i.e. for each compact L c Y there exist jo e Ν and c > 0 such that llsllz. < cqjo(goa), g£Sa . (j) By virtue of (i), if 4 is a closed subspace of Θ(Χ) (in the topology τχ), then Sa is closed in Θ(Υ) (in τγ). Moreover, for each compact L c Y there exist a compact Κ C X and a constant c > 0 such that I I s I I l <c||goa||tf, g e t " . (k) In the special case, if % is a closed subalgebra of Θ(Χ), then Sa is a closed subalgebra of Θ{Υ) and for each compact L c Y there exists a compact Κ c X such that I I S I I L < goaK, G E F . Indeed, by (j) we have ||g*||L < cgk ο or||a: for any g e Sa and k e N. Hence llglli, < cx /k g οα||ΛΓ· Letting k —• +oo, we see that we can take c = 1. (1) The estimate in (k) may be written in the form Lc^K)*", where for a compact set Η c Y and a family Τ C 0(F) we put HT := { y e Y : V f € T : | / ( y ) | < | | / | | f f } . The set Η is called the Τ-hull of H. The set H&iY) is called the holomorphic hull of Η. Τ-hulls will play a very important role in the characterization of 7"-extendibility — cf. § 1.10. Exercise 1.4.6. Prove that Remark 1.4.5(h) remains true for -8 c M°°{X) such that V/e4 VaeC: |a|>||/||* : J ^ € S. In particular, we can take S := J£°°,k (X), cf. Exercise 1.2.6(c).
- 43. 1.5 The boundary of a Riemann domain 29 1.5 The boundary of a Riemann domain Let (Χ, ρ), (Y, q) be Riemann domains over Cn and let φ: X — > Y be a morphism. =φ Our aim is to define an abstract boundary 3 X of X w.r.t. the morphism φ. The idea of such an abstract boundary is due to Grauert (cf. [Oka 1984], [Gra 1956], [Gra-Rem 1957], [Gra-Rem 1956], [Doc-Gra I960]). In the case where (X, p) = (G, id) (G is a domain in Cn ), (Y, q) = ( C , id), =id φ = id, the abstract boundary d G will coincide with the set of, so-called, prime ends of G. At first let us recall some facts and notions from topology. Let X be a topological space. We say that a non-empty family $ of subsets of X is a filter if: • Αι, a2 € y = » Αι η A2 G A non-empty family φ of non-empty subsets of X is said to be a. filter basis if: • VA1,A2e«p : A C Αι (Ί A2- It is clear that for each filter basis the family := {A c X : 3ße<p: Β c A} is a filter. We say that a filter # is convergent to a point a G X if each neighborhood of a belongs to We shortly write a G lim J. We say that a filter basis φ is convergent to α if α e lim (equivalently, each neighborhood of a contains an element of φ); we put lim := lim We say that a is an accumulation point of a filter $ (resp. filter basis φ ) if a G Ä for any A € # (resp. A G φ). Let us recall a few elementary properties of filters (cf. [Bou 1965]): • If # C & are filters and if a is an accumulation point of then α is an accumulation point of 5· • If a G lim & then a e lim for any filter D • If a is an accumulation point of J, then there exists a filter D $ such that a G LIMY'. • a G Ä iff there exists a filter basis φ consisting of subsets of A such that a G lim φ. • Let Y be another topological space and let φ: X — • Y. Then φ is continuous iff for any filter basis φ in X the filter basis :— {φ(Α): A G φ } satisfies the relation: ^(Ιίπιφ) c Ιίπι^(φ). • X is Hausdorff iff any filter in X converges to at most one point. If X is a Hausdorff space and lim # = {«}, then we write lim % = a. For a G X and A C X let Q5C(A, A) denote the family of all open connected neighborhoods U of a with U C A. Put <8C (A) : = <8C(A, X ) . Let us come back to the situation when (Χ, ρ), (Y, q) are Riemann domains over C" and φ: X — > Y is a morphism.
- 44. 30 1 Riemann domains Definition 1.5.1. We say that a filter basis α of subdomains of X is a φ-boundary point of X if: (1) α has no accumulation points in X, (2) there exists a point yo e Y such that lim φ(a) = yo, (3) for any V e Q3c(_yo) there exists exactly one connected component U =: e(a, V) of (p~HV) such that U e a, (4) for any U e α there exists a V e Q3c(jo) such that U = 6(α, V). =<p Let 3 X denote the set of all ^-boundary points of X. We put =φ =φ Χ := XU dx and we extend φ to φ: X —> Y by putting φ(a) := yo if a and yo are as above. =<p Moreover, w e put ρ : = q ο φ. In the special case when (Y,q) = (Cn , id) and φ = ρ we skip the superscript p = = =p =p and we write dX and X instead of 3 X and X , respectively. Note that: • If (Ύ, q) = (X, p) and φ = idx, then the «^-boundary is empty. • If G = E[ 0, 1), then 3 G consists of points from (3G)(0, 1] with'multiplicity' 1 and of points from (0, 1] with 'multiplicity' 2. Remark 1.5.2. If α satisfies (2), (3), and (4) and α has an accumulation point ;to € X, then lim α = *ο· Indeed, first observe that yo = lim<p(a) = φ(χο). Let U e 93c(*o) such that φυ is injective. Observe that U is a connected component of φ'~φ{ϋ))19) . Take V e 23c(yo) with V <ξ <p(U). We know that jc0 e V) =: A. Since A is connected, φ(Α) C <p(U), and ^o € A, we conclude that A c U. Thus U e which finishes the proof. Instead of filters, one can describe the -boundary using sequences of curves: Consider the family β of all sequences £ = C X such that: (a) (xk)kLi has no accumulation points in X, (b) there exists a yo ^ Y such that l i m ^ + 0 0 (p(xk) = yo> (c) for any neighborhood V of yo there exists a ko such that for any k, I > ko there exists a curve γ^ι: [0, 1] — > <p~l {V) such that η,ι(0) = Xk, n,/(l) = xi• Lemma 1.5.3. Let y = C X, (tk)kLi C (0,1), tk < tk+ι, tk —>· 1. Then the following conditions are equivalent: (i) (xk)fLi satisfies (b) and (c); We argue as in Remark 1.1.2(f): Let Uq be a connected component of (<p{U)) that contains U. Then ψ := ° <PuQ maps Uq into U and ψ — id on U. Consequently, by the identity principle, ψ — id on Uq, i.e. Uq = U.
- 45. 1.5 The boundary of a Riemann domain 31 (ii) there exists a curve γ: [0, 1) —• X such that y(tk) = Xk, k > 1, and the limit l i m ^ i φ ο γ (t) exists 20 Proof, (ii) =>• (i): Clearly, lim*->.+«>?(**) = lim*^+ 0 0 <p(y(tk)) =: yo- Let V be an arbitrary neighborhood of yo- Take ko 1 such that γ it) e V for t > t^. Then for any i > k >kolhe curve y l^,^] connects jc^ with χι in <p~l (V). (i) =>· (ii): Let ( V / ) ^ be a basis of neighborhoods of yo with V/+i C Vj, j > 1. For any j there exists a k(j) such that for any k > k(j) the points Xk, Xk+ can be connected in (p~l (Vj) by a curve (we may assume that k(j) < k(j + 1)). Let yj be a curve connecting the points xk(j), · · ·, Xk(j+) in ^c1 (Vy), j > 1, and let yo be an arbitrary curve connecting χ ι , . . . , xjt(i) in X. Now, we glue the curves step by step yo U γι U Y2 U . . . and, after an appropriate change of parametrization, we get a curve γ: [0, 1) —> X with γ(t^) = Xk, k > 1. It is clear that limf_+i φ °9(t) = yo· Π Remark 1.5.4. Let j: = C X be a sequence such that there exist [0, 1) 3 tk / 1 and a curve γ : [0, 1) —> X with the following properties: • xk = y(tk),k > 1, • dx(xk) —> 0, • y0 •'= lim*-*.+oo <p{xk) exists, φ the limit limf ^i ρ ο y (?) = zo e C" exists 2 l Then y 6 Θ. Indeed, by Lemma 1.5.3, it suffices to check that l i m ^ i φ ο γ (ί) = yo· Observe thatgo^oy = ρογ. Hence q(yo) — zo- Let V be an arbitrary univalent neighborhood of yo- Take ko such that (p(xk) e V for k > ko and ρ ο γ (ί) e q(V) for t > to := tk0. Consider the curves σι :— φ ο y |[>0,i): [ίο, 1) —• Υ, σ2 := (qvrl (p°Y[to,i))·· Ho, Ό —• V. Observe that σι (ίο) = &2(to) = <p(xk0) and Ι οσ ΐ = q οσ2 = ρ ο y |[ί0,ΐ). Hence, by the identity principle for liftings, σ = σ2. In particular, φ ο y (ί) g V for t > ίο- Remark 1.5.5. If (xk)^Lι satisfies (b) and (c) and has an accumulation point xo € X, then lim*_>.+00 Xk = xq (cf. Remark 1.5.2). Indeed, assume that limi^+oo Xkv — Let U e Q3c(xo) be such that φυ is injective. Note that U is a connected component of φ~λ (φ(ί/)). Let ko be such that for any k, I > ko the points λ:*, χι can be joint in φ~ι (φ(ϋ)) by a curve y*^. Now let v > o be such that kvo > ko and Xkv e U for ν > ν>ο· In particular, any point Xk with k > ko can be joint with x^ in φ~χ {φ{υ)). Hence Xk e U, k > fco· For £ = (xk)™=v l' = (x'k)?=l e Θ we write y ~ p' whenever: Equivalently: φ ο γ extends to a curve γ: [0, 1] —> Υ. 21 ') Equivalently: the curve ρ ο γ : [0, 1) —> C" extends continuously to [0, 1],
- 46. 32 1 Riemann domains • lim^+oo <p(xk) = linu_»+00 <p(x'k) =: • for any neighborhood V of yo there exists a ko such that for any k > ko the points Xk and x'k can be connected in <p~x (V) by a curve. Note that y is equivalent to each of its subsequences. Observe that is an equivalence relation. Put dX := Θ/~ SS ^ ψ SS and define φ : 3 Χ —> Υ, φ ([y]~) := lim^+oo <p(xk)· =φ χζφ Proposition 1.5.6. There exists a canonical bijection Ξ: 3 X —> 3 X such that « =<p φ ο Ξ = φ on 3 Χ. =φ Proof. Let a € d Χ. Let ( V * j be a basis of connected open neighborhoods of yo := <p(a) with Vjt+i C Vk, k > 1. For each k take an arbitrary point Xk e £4 := C(a, Vk) and let ρ := (xk)fLi· Obviously, 1ΐηι^+ 0 0 <p(xk) = yo- Observe that Uk+ι C Uk and, therefore, Xk can be connected with xg in <p~l (Vk0) for any k, I > ko. Thus y satisfies conditions (b) and (c) (from the definition of the family β). Suppose that (x k)kL has an accumulation point jco in X. Take an arbitrary U = β(α, V) e ο and let Vk0 C V. Then Uk0 C U and hence Xk € U for k > ko. In particular, xq e Ü. Thus jco is an accumulation point of a; contradiction. We have proved that y € 6 . Now let (VpJ^j be another basis of neighborhoods of yo and let y' = C ^ ) ^ be constructed with respect to {Vk)^=l (x'k e U'k := C(a, V£), k > 1). We will show that y ~ y'. Take an arbitrary neighborhood V g ®c(>o) and let Vk0 U V^ c V. Then Uk0 U U'k(j c C(o, V) and, consequently, for any k > ko the points Xk, x'k can be connected in β(α, V) C <p~l {V). =φ πφ Thus we can define a mapping Ξ: dX —> 3Χ, Ξ(α) := [y]~. Obviously « = =<p φ ο Ξ = φ on 3 Χ. We will prove that Ξ is injective. Suppose that Ξ (α) — Ξ(α'). In particular, φ(ο) = φ{α!) —. yo- Fix a basis (Vfc)^i of connected neighborhoods of yo with Vk+i C Vk,k> 1, and let xk e Uk := 6(α, V*), x'k € U'k := 6(α', Vk), k > 1, y := (xk)kLi* := W e know t h a t ? ~ ?'· T a k e an arbitrary U = e(a, V) and let ko be such that for any k > ko the points xk, x'k can be connected in (p~x {V). Let k > ko be such that Vk C V. Then Uk U U'k c U. Hence 6(α', V) = U and therefore U e a'. Thus α = α'. It remains to show that Ξ is suijective. Let y = (xk)kLi e β . Observe that for any connected neighborhood V of yo := linu^+oo (p(*k) there exists exactly one connected component, say Gy, of <p~l {V) such that ^ e Gv for fc » 1. It clear that Gy remains the same if we substitute y by an equivalent sequence. Obviously, if W C V, then Gw C Gv. Put o: = {Gv: V e ®c(yo)}· It is clear that α is a filter basis and that lim φ(a) = yo· We will show that ο has no accumulation points in X. Suppose that xq is an accumulation point of o. Then, by Remark 1.5.2, lim α = χο· In
- 47. 1.5 The boundary of a Riemann domain 33 particular, for any neighborhood U of xo there exists a V g ®c(yo) such that Gy C U. Consequently, Xk —> contradiction. Finally, directly from the definition, we get Ξ (α) = (recall that any subse- quence of ι is equivalent to j). • =<p Our next aim is to endow X with a Hausdorff topology which coincides with the initial topology on X and is such that the mapping φ is continuous. =φ Let α G 9 X. By an open neighborhood of the point α we will mean any set of the form Üa:=U U{b g 9X: U e fo}, where U g a. Lemma 1.5.7. (a) For a', a" G 9 X we have: α' φ a" iff there exist U' g a', U" g a" such that U' η U" = 0. =<p (b) The topology of X is Hausdorff. (c) The mapping φ is continuous. Proof, (a) The implication ' < = ' is trivial. For the proof of ' = » ' suppose that U' Π U" Φ 0 for any U' g a', U" g a". Let y'Q := lim^iaO, yß : = limφ(α"). Suppose that y'Q φ yß. Choose V' e ftciy'o, J'), V" G Y) such that V' n f = 0 and let U' := 6(α', V'), U" := 6(α", V"). Obviously, U' Π U" = 0 ; contradiction. Thus y'Q = Now take an arbitrary U' = 6(α', V). Let U" := G(a", V). Since U' Π U" φ 0 , we get U' = U" and therefore U' g a". Consequently, a' = a"; contradiction. =<p (b) Take a', a" g Χ , α! φ a". It suffices to consider the following two cases: • a' := a' e X, a" e d X: Since a' is not an accumulation point of a", there exist U' g X), U" € a" such that U' η U" = 0 . Then U' η = 0 . • α', α" e 9 X: By (a) there exist U' G a', U" G a" with U' Π U" = 0 . Suppose =<p that ύ'α,Γ)ϋ^„ φ 0. Then there exists ab G 9 X such that (/', U" e$b. Consequently, 0 = U' (Ί U" g fo; contradiction. =ψ ic) Let OG dx, yo := φ(α). Take V,V2 g <8c(>o, 1") with V2 <£ V and let U2 := C(o, Vi). Suppose that there exists a b e {Ui)a Π 9 X with ζ := <p(b) # V. Let V3 g Q3c(z, y) be such that V3 Π V2 = 0 and let U3 := 6(b, V3). Then 0 = U3 Π Ü2 G ; contradiction. This shows that <ρ((ί/2)α) C Vi- • The following continuation problem will appear several times in the sequel. Let Γ be a topological space, let S be a nowhere-dense subset of Τ, and let f:TS —• X
- 48. 34 1 Riemann domains be a continuous mapping such that φ ο f extends continuously to a mapping Τ —> Y. =<p Does the mapping / itself extend continuously to a mapping Τ —> X ? Proposition 1.5.8. Let Τ be a locally connected topological space and let S be a nowhere-dense subset of Τ such thatfor any domain D C Τ the set DS is connected. Let f : TS —>· X be a continuous mapping such that φ ο f extends to a continuous =<p mapping g: Τ —> Y. Then f extends to a continuous mapping f : Τ —> X . Observe that / is uniquely determined. Proof. Take to e T. Let _yo := g{to) and let V e 55c(}O)· Since g is continuous, there exists a D e QSc(io) such that g(D) c V. Recall that DS is connected. Hence f(D S) C <p~x (V) is also connected. Let Gy denote the connected component of <p~x {V) that contains f (D S). Observe that Gy is independent of the choice of D e ©c('o) with g(D) c V. Define o(/0) := {Gy: V e ®c(yo)}· It is clear that α(ίο) is a filter basis and lim <p(a(to)) — >>o. Moreover, for any V e 53c(yo) there exists exactly one connected component U of <p-1 (V) such that U € o(?o) (simply U := Gy). Note that if t0 i 5, Gh(yo r) = Ρ x ( f ( t 0 ) , r) for 0 < r < dx(f(t0)); in particular, lim α(ίο) = /(Λ))· There are the following two possibilities: =<p • α(ίο) has no accumulation points in X. Thena(io) G 3 X. We put /(ίο) := α(ίο)· • a(/o) has an accumulation point Jto e X. Then lim α(ίο) = (cf. Remark 1.5.2). In this case we identify α(ίο) with xq and we define /(ίο) = *ο· =φ = We have defined an extension / : Τ —> X of / such that φ ο / = g. It remains to prove that / is continuous. Fix a U = Gy0 € o(io) and let Do e Q3c(fo) be such that U is the connected component of φ~λ (Vo) that contains /(Do S). Obviously, U e a(t) for any t e Do- Hence for t e Do we get: • if a(f) € X, then f ( t ) e U; —ψ =< p • if a(io) e 9 X and a(t) g 3 X, then o(i) 6 Ua(to). Now we only need to observe that if Jto := α(?ο) € X, then a(t) e X for t e Do A provided U is small enough. In fact, let U := Ρχ(χο» Ό = r) with r := dx(xo), and let /(Do S) C U. Observe that / = (φυ)~1 ° g on Do S. Hence (^|[/)_ 1 (g(0) e Gy for any t e D0 and V e ®c(g(i))· Thus a(0 = C<p|c/)_1 Cff(0), t e D0. • =<P The following result characterizes the geometry of 3 X. =φ λ =φ Proposition 1.5.9. For any α € 3 X and for any neighborhood Ua C X there exists a neighborhood Wa c Ua such that άχ = du on W. In particular, lim dx(y) — 0. Xsy^f-a
- 49. 1.5 The boundary of a Riemann domain 35 Proof. Let yo : = <P(&) and suppose that U = C(o, V), where V € ?Bc(}O)· Fix 0 < r < dY(y0) such that ¥Y(yo, r) <s V and let Uj := β(α, FY(yo, r / j ) ) , j > 1. We have U D U D U2 There are the following two possibilities: 1°. There exists a jo such that y0 i <p(Uj0): Put W := U2j0. Take x0 € W. Suppose that άχ (XO) > |/?(JCO) — ^(YO)L- Recall that ρ = qocp. In particular, |p(*o) — ^(yo)! < r/(2yo). Hence we find an 5 such that |p(*o) — ^(>Ό)Ι < s < r/(2jo) and Ρχ(*ο. s) exists. Since <ρ(Ρχ(*ο, s)) = PY(<p(x0), s), we have: ^(Ργ(φ(χο), s)) = ρΦχ(χο, S)) C nq(yo), r/jo). Recall that <p(x0) e FY(yo, r/(2j0)). Consequently, Py (#>(*o), s) c FY0>o, r/jo) and € Py (<P(*o), s). This means that Ρχ(*ο, s) c Uj0 and that > > 0 e <p(Uj0)·, contradiction. Thus dx(x0) < p(xo) — q(yo)· By the same method as above we easily show that ΡχOo) C Ujo. Finally άχ(xo) = dUjQ(x0) = dv{x0). 2°. yo = φ(Χ)) with Xj e Uj for any j > 1: Put W := U4. Take xo e W and suppose that dx(x0) > r/2.^ Then F(q(y0), r/4) C ςΦγ(φ(χο),Γ/2)) C Cyq), 0· Hence Py(y0,r/4) C Ψγ(φ(χ0), r/2) C ΡY (yo,r) and consequently, t/4 C Ργ(χο, r/2) C U. In particular, Xj = X4 for j > 4. This means that X4 is an accumulation point of a; contradiction. A ThusdxCxo) < r/2 and, consequently, Ρχ(^ο) dU. Finally άχ (*o) = djx (xo) = du(xo)· • Remark 1.5.10. Let Ω c X be a domain and let ψ := φω 22 Consider the ψ- boundary of the domain (Ω, ρ|Ω)· We will show that there exists a canonical continuous mapping =ψ =φ Τ : Ω —• Χ such that Ύ ( Ϋ Ώ ) C (9χΩ) U (9 Χ) 23) and φ ο Τ = ψ. =Ψ Take ο € 9 Ω. For any U e α, let V c Y be a connected neighborhood of yo := ψ (a) such that U is a connected component of js~l(V). Let G u y denote the connected component of <p~l(V) that contains U. Define T(a) := {G(j, ν • U e a, U is a connected component of φ~χ (V)}. Then Ύ(α) is a filter basis of domains in X and lim φ(Ύ(α)) = yo· If Τ (a) has no =φ accumulation points in X, then Τ (a) G 9 X. Suppose that xq e X is an accumulation point of T(o). Then φ(χο) = YO and JCO = lim Τ(α) (cf. Remark 1.5.2). Observe that xo G 9χΩ. Moreover, we put Τ (a) := a for a € Ω. Note that ψ: (Ω, ρ|ω) —*• (Υ, q) is a moφhisIn. 3χ Ω denotes the standard boundary of Ω (in the topology of X).
- 50. 36 1 Riemann domains =ζψ =φ =ψ Thus we have defined the mapping Τ: Ω —> X such that Τ ( 3 Ω ) c (3χΩ) U (3 X) and φ ο Τ = ψ. Since T(Üa) C (Gu,v)r(a)> the mapping Τ is continuous. *<P =Ψ *φ =φ Now we would like to determine a 'maximal' domainX c X such that (Χ , ρ χ is a Riemann domain. The idea is to take *<p =<p X = XU ('thin' parts of 3 X). A set Ρ C X is called thin if for any point a e X there exist U e and / € 0(U), f ψ 0, suchthat Ρ Π U C /_ 1 (0). Let &(X) denote the class of all closed thin subsets of X. Proposition 1.5.11. Let (Χ, ρ), (Υ, q) be Riemann domains over C" and let <p:(X,p)—+(Y,q) be a morphism. =<p =<p (a) Assume that α G 3 X is such that there exists a neighborhood U C X of a with thefollowing properties: • V := <p(U) is open in Y, • Ρ := φ(υ (Ί 9 Χ) 6 Ä(V), =φ • φ: U 3 Χ —> V Ρ is biholomorphic. Then the mapping φυ • U —> V is homeomorphic. =<p (b) Let Σ denote the set of all points a e 3 X which satisfies the above conditions. Put *<p X := X U Σ. Then: *<P =φ • (Χ , ρ I *Ψ) is α Riemann domain over Cn , χ = *φ =φ • φ *φ (Χ , ρ I *φ) —> (Κ, q) is a morphism, χ χ *φ • Σ < ε Α(Χ ). * *ρ In the case (Υ, q) = (Cn , id), φ = ρ, we will writeX instead of X . =<p Proof, (a) Put UQ:— U DX. Let us start with the following remark. =<p = (*) Suppose that α e U Π 3 X, U' e a, yo := φ{α), V' e ®c(;yo)· Then there exists a U" = e(o, V") such that U" CU'DUq and V" c V Π V. _ Indeed, let U' = C(a, V) and let U2 = e(o, V2) be such that (U2)A C U. Let V" G ®c(yo, y'n Vi Π V2nV) and putt/" := e(a, V"). Notethat U'NU2RU" Φ 0. Since <p(U") C V Π V2, we get U" C U' DU2CU' η UQ.
- 51. 1.5 The boundary of a Riemann domain 37 First we prove that φ is injective on U. Suppose that there are a', a" e. U such that α' Φ a" and φ(ο!) = φ{α") =: yo- =φ We may assume that a a" e U Π 3 X. By Lemma 1.5.7(a) there exist U e a', U" g a" with U' η υ" = 0. By (*) we may assume that U' = 6(α', V), I/" = 6(α", V) with U' U U" c i/o and V' C V. Observe that V Ρ is connected and therefore 0 φ ( φ υ ϋ Γ χ { ν ' Ρ ) c U' Π U" contradiction. Now we prove that {<pu)~x is continuous. Take a e f / f l dX and let yo φ(ο). Take Vι e V) and let U := C(a, Vi). By (*) U η U0 φ 0. Hence (Plc/0)_1 (y i C C/i. We will prove that (^|[/)_1 (Vi) C ((7ί)α· We have to show =φ that if b € ί/ Π 3 X is such that <p(b) g Vi , then U €$B· Indeed, by (*) there exists a u2 = e(b, y2)with U2 C Uq and C V. Observe that (Pli/0)_1 (V2Ρ) C Hence U2 C U, which shows that U € *φ (b) First, observe that if α 6 Σ and U is as in (a), then U C X . Consequently, *φ =φ _ X is open in X . By (a), the mapping φ*Ψ is locally homeomorphic. Recall that χ —φ = =φ =φ ρ = q ο φ. Hence ρ *Ψ is locally homeomorphic and therefore (Χ , ρ is a χ χ = *Ψ =φ Riemann region over C". Obviously, φ*φ: (Χ , ρ *φ) —(Y, q) is a morphism. χ χ Observe that if U, V, Ρ are as in (a), then ψυ maps biholomorphically Σ Π U onto *φ *φ *φ P. Thus Σ € Ä(X ). In particular, X is dense in X , which implies that X must be connected. • Example 1.5.12. Consider the following elementary example. Let(X, ρ) (E*, id), (Y, q) = (C, id), φ := id. Then: Ε^ coincides with the topological boundary dE U {0} of coincides (as a topological space) with E, Σ = {0} andi* = E. *(ρ The following proposition will show that X is in some sense maximal. Proposition 1.5.13. Suppose that W C X is an open subset such that: • <p(W) = V P, where V is an open subset ofY and Ρ e £ ( V ) , • φ: W —> V Ρ is biholomorphic. *<p _ Then there exists an open set U C X such that W C U and φ: U —> V is biholomorphic. Proof. Let Σι denote the set of all points χ e X such that • y : = φ(χ) € Ρ , • there exist W € ©c(jc), V7 e ©c(y, V) such that <pw>: W' —> V' is biholo- moφhic,
- 52. 38 1 Riemann domains • W w r k v ^ p j c w . Observe that W U Σ] is an open subset of Χ, Σι c W, and that int Σι = 0 . Put ψ := <pw. For y e P0 := Ρ let a(y) := { V ( V ) : V' G V)}, where V ( V ) denotes the connected component of φ~] (V') that contains the set ( V'). Notice that PQ is a relatively closed subset of P. Step 1°. a(z) e dX, ζ e P0: Fix a yo G PQ. It is clear that a(yo) is a filter basis satisfying conditions (2), (3), (4) of Definition 1.5.1 (lim qp(a(;yo)) = Jo)· Suppose that a(yo) has an accumulation point xo G X. Obviously, φ(χο) = yo· Observe that G W. Otherwise, we would find W,W2 e Q3c(jc0), V,V2 G ®cOo, V) such that V2 <£ V, Wi Π W = 0 , : Wi —> V is biholomoφhic, and = V2. Put U2 := V(V2). Observe that W3 := f ~ V 2 P) C U2. Since W2 Π U2 φ 0 (jcq is an accumulation point of a(;yo)), we get W2 c U2. Fix an χι e W3 and connect this point with xo inside U2 by a curve γ. Then φ ο γ lies inside V2 although γ crosses W2 contradiction. Now choose W' e V' e 95c(yo, V) such that <pw> W' — • V is biholomorphic. We will show that (φν')~Χ (ν> P ) <ZW. Fix an x' e W' Π W and suppose there exists an x" e W' W with φ{χ") G V' P. Connect x' with x" by a curve γ inside W' (φψ')~Χ (Υ' Π Ρ). Then φ ο γ is a curve inside V P. So ο {φ ο γ) is well defined. By the identity principle for liftings both curves are equal. In particular, χ" = γ (1) g W contradiction. This shows that jco € Σι; contradiction. Define U := WU Σι U {α(ζ) : ζ e P0}· We claim that U satisfies all the required conditions. Step 2°. φ: U —> V is bijective: The suijectivity is clear. Take x', x" e U, χ' ψ χ " , such that ψ ( χ ' ) = <p{x") =: yo· The only case which is not obvious is that when χ', x" g Σι . Then there are disjoint neighborhoods W' e y$c{x') and W" e ^ d x " ) and V' e ®c(>>o, V) such that φ maps biholomorphically W onto V' and W" onto V'. Take a point y G V' P. Then y has two preimages in W; contradiction. *φ Step 3°. U is an open subset of X : *<p First observe that U c X • Indeed, the set PQ as a relatively closed subset of Ρ is =<p thin and the mapping Φ: U dX —• V PQ is biholomorphic. =<p Thus, it suffices to show that U is an open subset of X .
- 53. 1.6 Union, intersection, and direct limit of Riemann domains 39 Let a(jo) be one of our filter basis with yo e Po. Choose V, V2 G ®c(>o, V), V2 V], and let U :— V(Vi). Take as a neighborhood of a(jo) the following set u '·= Π (Ü)a(y0)· We will prove that U C U. First we prove that i / i C W U E , = : W. Otherwise, we can find points x' G U Π W, x" G U W. Let γ be a curve in U1 connecting x' and x". Let ίο be the first parameter such that y (ίο) G 3 W. Put x* := y(to). Then χ* £ Σι. It is clear that χ* G 3W. Observe that φ ο γ is a curve in V starting at the point φ{χ') G V P. Suppose that y* := φ{χ*) £ P. Choose W3 G ®c(jc*) and V3 e V P) such that <p|w3: — • V3 is biholomoφhic. Then, since W3 Π W Φ 0 , we get W3 C W (use the identity principle for liftings); contradiction. So we know that y* e Ρ Π V. Now, as in Step 1°, we show that x* e Σι; contradiction. Hence we know that Ui c W. =<p Take α e U Π 3 X. We want to show that α e U. Put y' := φ{α). Of course, then / e V2 <£ V. Observe that y' e Ρ Π V2. For, otherwise y' e V P. Hence there exists exactly one point w' e W with φ{ν') = y'. We claim that in such a case w' must be an accumulation point of a. For, fix a U' e a. Let W' e W) and U" = e(o, <p{W')). Observe that if U" Π W φ 0 , then U" = W'. Recall that U e j f l . We already know that U C W. Consequently, U" Π W Φ 0 . Since int Σι = 0, we get U" Π W Φ 0 and hence U" = W'. In particular, U'nW' φ 0 . Assuming that yr € φ(Σ,) also leads to a contradiction. So we start with the information that <p(A) =: y' € Pq. We claim now that α = α(/). Otherwise there are U' e α and U" = V( V") e a(y') with U' η U" = 0 . We may assume (cf. property (*) from the proof of Proposition 1.5.11) that U' = C(o, V") and that V" C In particular, U' <ZU. Hence U' = U"; contradiction. The proof of Proposition 1.5.13 is completed. • 1.6 Union, intersection, and direct limit of Riemann domains The union of Riemann domains. Let (X, p) be a Riemann domain over C" and let 0 φ £ c Θ(Χ). Consider a family α,·: (Χ, ρ) —> (F,, qi), i G I, of -^-extensions. We would like to define an -ί-extension a : (Χ, ρ) —> (Υ, q) which behaves like the union of the above extensions (cf. [Gra-Fri 1976], [Ohg 1979]). Definition 1.6.1. We say that S weakly separates points in X if for any x,x2 G X with jci Φ JC2 and p{x) = p(x2) there exist f € S and ν G Z" such that 3 ( j c 1) Φ
- 54. 40 1 Riemann domains dv f(x2); equivalently, for any χι,χ2 € X with x φ χι there exists an f e S such that TXl f φ TX2f (as formal power series 24 ^). Observe that the condition is empty if X is univalent. In particular, if G is a domain in Cn , then any family <8 weakly separates points in G. If S is 3-stable, then the weak separation means that the family -8 separates points in stalks. Consequently, if -8 is 3-stable and ρ e <$n , then -8 weakly separates points in X iff it separates points in X. We say that an ^-extension α: (X, p) —> (F, q) satisfies condition (S) if the family <8a weakly separates points in Y. Proposition 1.6.2. There exists an -8-extension a: (Χ, ρ) —> (y, q) satisfying (S) such that: • for any i e I there exists a morphism <pt: (7,·, qi) —> (7, q) with φι ο αι = a, •U ieJ<Pi(Yi) = r, • if β: (Χ, ρ) —> (Ζ, r) is an -8-extension satisfying (S) such that there exist morphisms ψι: (7( , qi) —> (Z, r), i e I, with ψι ο α,· — β, i e I, then there exists a morphism σ : (Κ, q) —> (Ζ, r) such that σ οφι = ψί, i e I (inparticular, σοα = β). Moreover, the extension a: (X, p) —> (Y, q) satisfying the above conditions is uniquely determined up to an isomorphism. Observe that the morphisms φι: (Yi,qi) —• (Y,q), i € /, are uniquely deter- mined. We write (a: (X, p) —-> (Y, q)) = J(ai: (X, p) —• (Yi,qi)) iel and we say that a: (X, p) —> (F, q) is the union of the family cn:(x, p)—>(Yi,qi), i e I. Proof For any i e I and / e S, let /,· e &{Yi) be such that /,· ο α,· = / (i.e. fi := /«<)· Let f : = | J r / x{i}. For (ji, i'i), (y2, h) e f we put (yi.ii) ~ (yi, 12) :«=> Tylfil = Ty2fi2, f e <8; (in particular, qii (yj) = qi2(yi) and fh (yi) = fi2(yi) for any / e -8). We define q:Y-+ Cn , q([(y,i)]~):=qi(y), f : Y ^ C , f[(y, i)]~) := My), f € 4, ψί: Yi Y, <pi(y) := [(y, i)]~, i e I. 24 ) Recall that two formal power series Σν<ζζ"+ av(z — a)v , bv(z — b)v are equal if a = b and av - bv for any ν e Z!j_.
- 55. Random documents with unrelated content Scribd suggests to you:
- 56. Wood sorrel, convex -Nov. 23. Oxalis convexula. Wood sor., la. flow. -Nov. 21. Oxalis grandiflora. Wood sorrel, lin. -Nov. 26. Oxalis linearis. Wood sor., lupin-lea. -Nov. 27. Oxalis lupinifolia. Wood sor., three-col. -Nov. 30. Oxalis tricolor. Wood sor., trump. fl. -Nov. 22. Oxalis ubiflora. Yarrow -Oct. 16. Achillæs millefolium. Yew -Nov. 6. Taxus baccata. Yew tree -Jan. 13. Taxus baccata. Zinnia, elegant -Aug. 14. Zinnia elegans. V. CORRESPONDENTS’ INDEX.
- 57. Amicus, 718. Bees and birds, 647. Cantab, 697. Constant Reader, 518, 1356. Causidicus, 517. C. L., 965, 1358, 1385. C. R. H., 1331. Δ, 658. D., 1063, 1122. D. G., 466. Dorsetshire gentleman, 206. Easter articles, 416, 519. E. J. C., 717, 944, 1337. Foster, John, 1573. Friend, a, “Item,” &c. 238. G. *, 1320. Gertrude Grizenhoofe, 1375. Gwilym Sais, 1421. H., 1454. H. C. G., 719. Hertfordshire letter, 565. H. H., 1566. H. M., 1343. H. T. B., 562. Jack Larking, 289. J. B., 244, 426, 799, 1077. J. H. H., 1575. Jibb, Joseph, 1482. Johnson, John, 1498. J. N., 1487. J. S., 802. Lector, 382, 727, 1124. Leeming, Joseph, 1467. Licensed Victualler, 1253. L. S., 425, 431. May-day Cow, 571.
- 58. Native of Penzance, 561. —— of St. Catharine’s, 1405. Naturalist, 614. Nicolas, Mr. N. H., 416. North Briton, 1518. O. F. S., 1015. P., Mr., 244. Pompey, 944. Prior, J. R., 144. R. N. B., 242. R. R., 1501, 1508. S. G., 1436. S. G. *, 1439. Sheffield letter, 591. Sigma, 841. S. R., 430, 1011, 1287. S. W., 253. T. A., 421. Tim Tims, 1308. T. N., 645, 898, 1080, 1606. T. O., 1580. Twelfth of August, 1099. Twenty-ninth of February, 597. Wentana Civis, 1379. W. G. T., 1510. ¶ ¶, 525. VI. INDEX TO THE ONE HUNDRED AND SEVENTY ENGRAVINGS CONTAINED IN THE VOLUME. 1. Ærial, the, 1455. 2. Amelia, princess, her autograph, 1076.
- 59. 3. Apostle spoons, 178. 4. April, 407. 5. —— Fool, 410. 6. Aquarius, 141. 7. Aries, 375. 8. August, 1058. 9. Autumn, 1282. 10. Barber, 1254. 11. —— ancient, 1266. 12. ——’s basin, 1256. 13. —— candlestick, 1255. 14. —— chafer, 1256. 15. —— chafing-dish, 1257. 16. —— crisping-irons, 1257. 17. Barrow-woman, 903. 18. Bartholomew fair, 1223. 19. ——, 1226. 20. Bastile destroyed, 935. 21. Bear taking in Russia, 182. 22. Beard, cathedral, 1258. 23. —— Pick-a-devant, 1258. 24. Beaton’s, cardinal, house, Edinburgh, 711. 25. Boar’s head at Christmas, 1619. 26. Bona Dea, Frontispiece, 1655. 27. Boor’s head, 1622. 28. Boy bishop, 1559. 29. Bungay Watchman, 1627. 30. Burmese state carriage, 1522. 31. —— Tee, 1528. 32. Butler, Jacob, 1303. 33. Buy a broom, 807. 34. Calabrian minstrels, 1594.
- 60. 35. Canonbury tower, 634. 36. Card-playing, by children, 90. 37. Cats’ concert, 1106. 38. Chad’s well, inscription, 323. 39. Church of St. John, Clerkenwell, 1475. 40. Copenhagen-house, 858. 41. Cowper’s summer-house, 522. 42. Cressets, four. 43. Curfew, 244. 44. December, 1543. 45. Fantoccini, 1114. 46. February, 195. 47. Flamsteed’s horoscope, 1093. 48. —— autograph, 1097. 49. Flight of the Holy Family, 1650. 50. Flowers with symbols, 195. 51. Fountain at Tottenham, 1041. 52. Garrick’s autograph, 327. 53. ———— signature, 330. 54. Gordon, Jemmy, 698. 55. Grose, Francis, sleeping, 655. 56. ———— standing, 656. 57. Guy Fawkes day, 1432. 58. Gymnastics for youth, 19. 59. ——, Voelker’s, 1322. 60. Hagbush-lane cottage, 374. 61. Hair-dress, ladies’, 1261. 62. —— bull-head, 1261. 63. —— curls on wires, 1261. 64. Halifax gibbet, 147. 65. Hare and tabor, 1210.
- 61. 66. Heading-block and maul, 149. 67. Heart breaker, 217. 68. Hen threshing, 247. 69. —— speaking, 250. 70. Henry IX., K. of England, 33. 71. ———— reverse of his medal, 34. 72. Hipson, Miss, a dwarf and a Malay, 1174. 73. Hornsey Wood house, 759. 74. —————— lake, 762. 75. Huxter, 1214. 76. Hyde Park gate, sale, 1358. 77. Italian minstrels in London, 1630. 78. January, 1. 79. Joan of Arc’s fountain, 730. 80. John, St., at Patmos, 618. 81. July, 890. 82. June, 738. 83. King’s arms, a showman’s wood-cut, 1176. 84. Labre, B. J., 472. 85. Lamp, old, 833. 86. Lifting at Easter, 423. 87. Lion bait at Warwick, 986. 88. Little man, 1190. 89. Living skeleton, front, 1018. 90. ———————, profile, 1033. 91. ———————, back, 1034. 92. London insignia, 1442. 93. March, 311. 94. May, 538. 95. May-day at Hitchin, 567.
- 62. 96. ———— chimney sweepers, 583. 97. ———— milkmaid’s garland, 570. 98. Mermaid, a showman’s wood-cut, 1193. 99. Michael Angelo Buonarroti, 271. 100. Mid-Lent sport, 358. 101. Midsummer-eve bonfire, 823. 102. Nativity, the, 1610. 103. Nero and his senate, 458. 104. New London Bridge, 775. 105. November, 1418. 106. Octavia’s triumph, 458. 107. October, 1346. 108. Palm Sunday procession, 392. 109. Passion flower, 770. 110. Peerless-pool, 970. 111. —————— fish pond, 975. 112. Piper, the, 1626. 113. Pisces, 282. 114. Plough Monday sports, 71. 115. Porter, 1215. 116. ———’s part, 1216. 117. Printing office, 1134. 118. Pulpit, 839. 119. Richmond, Surrey, 602. 120. Sadler’s Wells’ angling, 343. 121. St. Anne and St. Joachim, 1010. 122. St. Bride’s Church, Fleet-street, 87. 123. St. Catharine, 1506. 124. St. Cecilia, 1495. 125. St. Crispin and St. Crispinian, 1395.
- 63. 126. St. Denys, 1370. 127. St. Dunstan and the Devil, 671. 128. St. George, 498. 129. St. Ignatius Loyola, 1050. 130. St. Michael and other archangels, 1328. 131. St. Nicholas, 1556. 132. St. Roche, 1120. 133. Sandal, ancient, 514. 134. September, 1146. 135. Shoe, ladies, 516. 136. Silenus, 450. 137. Simeon, St., Stylites, 35. 138. Sirius, 897. 139. Sluice-house, Hornsey, 695. 140. Somers’ Town miracle, 474. 141. Spring, 335. 142. —— dress, 14th cent., 337. 143. Squirrel, musical notes, two, 1366. 144. Starkey, capt., 922. 145. Stoning Jews in Lent, 295. 146. Summer, 818. 147. Summer dress, 14th cent., 819. 148. Sun and Earth at Midsummer, 378. 149. —— at Midwinter, 59. 150. Swallow, hirundo rustica, 506. 151. Temptation of St. Antony, 114. 152. Tiddy Doll, 575. 153. ——’s musical notes, 578. 154. Tree of Common Law, 234. 155. Twelfth-day in London, 47. 156. Valentine, postman, 215. 157. Virgin, Mater Dei, 1273.
- 64. 158. Want, Hannah, 1352. 159. Westminster-hall with its shops, 154. 160. Whitehead, a giant boy, 1195. 161. Wigs, travelling, 1260. 162. —— long perriwig, 1260. 163. —— peruke, 1259. 164. Wild-fowl shooting in France, 1575. 165. —— shooter’s hut, 1578. 166. Winter, 1560.
- 65. END OF VOL. 1.
- 66. Transcriber’s Notes General remarks This e-text follows the text of the original printed work. Unusual spelling and inconsistencies have been retained; French and German accents and diacriticals have not been added, except as mentioned below. The printed book has two column numbers per page, but not all text was printed in columns. This e-text therefore uses the two column numbers per page as page numbers. Depending on the hard- and software used and their settings, some characters or other elements may not display as intended. Elements of the text that were printed in Blackletter in the source document are displayed as Blackletter in this text. Several references are not present in the book; these have not been linked. Where single references point to multiple pages, these references have not been linked either. The hierarchy in headings in the original work is not always clear; for this e-text, months have been taken as chapters, dates as sections, and other headings as (sub-)sub-sections. The original work has several gaps where numbers are missing; these are represented here as blanks (as for example in Cyder, at per quart). In the Indexes, V precedes U. Statements about the scale of illustrations compared to the actual size of the object depicted may not be valid for this e-text.
- 67. Volume II and Volume III are available at Project Gutenberg as well (www.gutenberg.org). Specific remarks Page 41/42, ... long past away: copied verbatim from the Athenæum, not changed. Page 235, tailor’s bill: the errors in the calculations (family visit to theatre) have not been corrected. Page 415, Moveable feasts: the corrections provided have not been corrected in the preceding text. Page 515, left foot: should be right foot, or the illustration has been reversed. Page 532: shirt/shift: both are articles of clothing, but one of them is likely to be a typographical error. Page 652, “we may advise early rising ...: the quote may end either before or (more likely) after Milton’s poem; the closing quote is missing. Page 764, chose the part of genuine greatness.”: the opening quote is missing. Page 922, footnote: some sources give Groat Market, others give Great Market as Hall’s address. Page 931/932, poem by Hone: the quote closing ... for the “love ... is missing, and should probably be inserted after ... love ... or after ... Every-Day Book. Page 1055, ... whilst St. Ignatius was living.”: it is not clear where the starting quotes should go. Page 1091, ... because I could: the sentence is incomplete in the printed work. Page 1245, four successive years, from 1779 to 1780: as printed in source. Page 1415: ... ringing of bels ... ringing of bells ...: as printed in the original work. Page 1439: To the Aldermen of the Ward of: presumably the original precept would have had a space for the
- 68. name of the ward (cf. the second precept); the printed book does not show such space. Page 1568, St. Nicholas in Russia: the article has no relation to Russia. Page 1622: the original work has a single footnote with two footnote anchors. It has been assumed that the footnote applies to both anchors. Page 1692, first of April, 1811: it is not clear to what this entry refers. Page 1707 ff, Floral index: some plants are listed out of alphabetical order, this has not been corrected. Changes and corrections made Minor obvious punctuation (mainly missing end of sentence periods and periods after abbreviations), capitalisation and typographical errors have been corrected silently. Missing quote marks have been added silently where there was no doubt where they should go; otherwise they are mentioned above or below; excess quote marks have been deleted silently when there was clearly no need for them, otherwise they are mentioned above or below. Multiple footnote anchors for a single footnote: the footnotes have been copied as necessary. Footnotes have been moved to the end of the day to which they belong. Ibid. has been replaced with the actual title when footnotes have been moved apart, the actual title has been replaced with Ibid. when footnotes have been moved together. The lay-out of the lists of saints immediately under the date has been standardised. The lay-out of the Floral Directories has been standardised.
- 69. A.D./A. D. and B.C./B. C. have been standardised to A. D. and B. C., respectively. (Scottish) names M’... and M‘... have been standardised to M‘.... Page 9: “ added before to forsake ... Page 12: pubic-house changed to public-house Page 29: ” added after ... (ignorant) a schoolmaster. Page 39: Woordenbock changed to Woordenboek Page 46: question mark after Peratoras deleted Page 58: ... they use to set up ... changed to ... they used to set up ... Page 78: ... “as the earliest flower ... changed to ... as “the earliest flower ... Page 102: Sts. Felix changed to St. Felix (only one St. Felix has his feast on 14 January) Page 133: ... when he sees buds ... changed to when she sees buds ... Page 136: Keates changed to Keats Page 139/140: ” inserted after ... happy speed.— Page 188: me army changed to the army Page 197: mensis plancentarum changed to mensis placentarum; “food,” or cakes.” changed to “food,” or “cakes.” Page 200: ” deleted after ... the hands of the faithful. Page 208: .. that to the, ... changed to ... that to thee, ... Page 218: ” added after ... her pow’r displays. Page 257: Neogeorgus changed to Naogeorgus Page 305: Geshiete der Erfindungen changed to Geschichte der Erfindungen Page 322: Album Porrum changed to Allium Porrum Page 331: conge d’élire changed to congé d’élire Page 378: ” deleted after ... hosen, &c. Page 392: ” added after ... the lustre of his miracles, Page 405: “ inserted before it preserves the house ...
- 70. Page 412: un poisson b Avril changed to un poisson d’Avril Page 435: setting up off poles changed to setting up of poles Page 446: an Eastern Tale changed to an Easter Tale Page 465/466: ” added after ... of half the year to rise. Page 468: ” added after ... kind of monastery, Page 469: ” deleted after ... learned the Gregorian chant. Page 478: in the tower changed to in the Tower Page 503: rejoicing peels changed to rejoicing peals Page 507: Hirundo vrbica changed to Hirundo urbica Page 507: Hirundines vrbicae changed to Hirundines urbicae Page 529: Ferara changed to Ferrara Page 547/548: the bag-pipes straines changed to the bag-pipe’s straines Page 609: blow of tulips changed to bowl of tulips Page 621: ... died, in 1721 changed to ...died, in 721 Page 628: ‘ inserted before I would not,’ says I; ... Page 645: ‘We walked in the evening,’ says Boswell, ‘in Greenwich-park. ... changed to “We walked in the evening,” says Boswell, “in Greenwich-park. ... Page 655: This gentlemen ... changed to This gentleman Page 666: ” inserted after ... the Horticultural Society. Page 705: St. Marttin’s-in-the-fields changed to St. Martin’s-in-the-fields Page 710: Irid Lurida changed to Iris Lurida Page 754: ... ancient performances is ... changed to ... ancient performances are ...; ... numerous quickly plied- hammers ... changed to ... numerous quickly-plied hammers ... Page 786: S. R. S. changed to F. R. S. Page 800: ... I fall too ... changed to ... I fall to ... Page 802: ) added after ... with sparkling eyes, Page 832: ” added after ... Ile be thy Ward.
- 71. Page 834: ... all other things that be suspected,’ changed to ... all other things that be suspected,” Page 836: closing quote added after ... camell, Page 871/872, ... or. demurrer ... changed to or. demurrer; Sr. Tho. Hoby changed to Sr. Tho. Hoby Page 884: Shall, be himself destroyed at last changed to Shall be himself destroyed at last Page 885: ... took out station ... changed to ... took our station ... Page 901, Like friendship clinging: indented like other stanzas’ last lines. Page 905, ... amber-berries?” changed to ... amber- berries!” Page 929/930: QUATORZIANS changed to QUATORZAINS Page 932: ” added after ... for the “love Page 933: St. laus changed to St. Idus Page 950: ... readers patience ... changed to ... reader’s patience ... Page 956: Said I not true’ changed to Said I not true, Page 972: ” added after ... that once Perilous Pond, Page 975: ... Philosophical Transactions” changed to ... “Philosophical Transactions” Page 992: 3. inserted before Tiger, ... Page 1007: re-reformation changed to reformation Page 1022: shoulder-blade changed to shoulder-blades Page 1025: dorsal vertebra changed to dorsal vertebræ Page 1044: ” added after ... sound and visual display. Page 1051: ” added after ... t’other shoe on Page 1091/1097: ye, ye etc. standardised to ye Page 1102: by K. d. b. k. denotes ... changed to by K. b. d. k. denotes ... Page 1120: “ inserted before Sound as a roach.” Page 1123: ... the rev. Mr. G. —, changed to the rev. Mr. G—,
- 72. Page 1125, footnote [262]: “ inserted before till they be red ... Page 1133: ... diocess of Utrecht ... changed to diocese of Utrecht ... Page 1138: ” added after “Academy of Armory Page 1170: ... were the toyseller’s; ... changed to ... were the toysellers’; ... Page 1201: Win-the fight changed to Win-the-fight; Zeal-of the-land changed to Zeal-of-the-land Page 1216: ” added after “fancy monger Page 1218: ” added after ... in its prime. Page 1269: ” inserted after ... brought in the brush. Page 1285: ... and ourang-outang ... changed to ... an ourang-outang ... Page 1286: Anderlent changed to Anderlecht Page 1295: the town goal changed to the town gaol Page 1309: Roussins dé Arcadie changed to “Roussins d’Arcadie” Page 1325: A. 409 changed to A. D. 409 Page 1336, row Stationary wares, ...: 0 added in column for pence Page 1359: he bytes; not he is no fugitive changed to he bytes not; he is no fugitive Page 1362: Faith changed to St. Faith Page 1368: ’ added after ... jest and fancy. Page 1373: October 1. changed to October 11. Page 1379: ” added after Employ thy precious hours. Page 1381/1382, footnote [352]: the footnote anchor was not present in the printed work; it has been inserted at what seemed the most likely place. Page 1416: ” added after ... a blessing never ceaseth.; Corcopsis ferulefolia changed to Coreopsis ferulifolia Page 1421/1422: ” added after ... which are her winding sheet. Page 1430: ” added after ... please to remember Guy. Page 1459: ” added after ... valuable consideration,
- 73. Page 1470: ” added after ... the altar to be removed. Page 1493: header FLORAL DIRECTORY inserted (November 19) Page 1499/1500: ... and do so no more.’ changed to ... and do so no more.” Page 1504: ” added after ... might be torn to pieces. Page 1547: 52′ changed to 52° Page 1551: ” added after ... the garter of the bride. Page 1600: Ant. a Wood changed to Ant. à Wood (elsewhere referred to as Wood); ... between 1546 and 1552: changed to ... between 1546 and 1552. Page 1654: ” added after ... Eve of New Year’s-day, Page 1665: Carracioli changed to Caraccioli Page 1670: 645 changed to 646 (Dotterel catching) Page 1672: 1809 changed to 1089 (Flamsteed) Page 1675: 6 6 changed to 656 (Grose) Page 1680: quatorzians changed to quatorzains Page 1692: .. for 1825; changed to ... for 1825 (Sculpture and painting) Page 1695: Stroud changed to Strood (Tail-sticking) Page 1697: 1600, 1160 changed to 1160, 1601 (Upcott) Page 1705: 875 changed to 877 (Hagbush-lane); 992 changed to 922 (Captain Starkey) Page 1720: 159. changed to 149. Page 1726: 33. changed to 337.
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