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Complex Manifolds 1, reprint with errata Edition James
Morrow Digital Instant Download
Author(s): James Morrow, Kunihiko Kodaira
ISBN(s): 9780821840559, 082184055X
Edition: 1, reprint with errata
File Details: PDF, 8.07 MB
Year: 2006
Language: english

COMPLEX MANIFOLDS
JAMES MORROW
KUNIHIKO I(oDAIRA
AMS CHELSEA PUBLISHING
American Mathematical Society· Providence, Rhode Island

2000 Mathematics Subject Classification. Primary 32Qxx.
Library of Congress Cataloging-in-Publication Data
Morrow, James A., 1941-
Complex manifolds / James Morrow, Kunihiko Kodaira.
p. cm.
Originally published: New York: Holt, Rinehart and Winston, 1971.
Includes bibliographical references and index.
ISBN 0-8218-4055-X (alk. paper)
1. Complex manifolds. I. Kodaira, Kunihiko, 1915- II. Title.
QA331.M82 2005
515'.946---dc22
© 1971 held by the American Mathematical Society.
20051
Reprinted with errata by the American Mathematical Society, 2006
Printed in the United States of America.
§ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10987654321 11 10 09 08 07 06

Preface
The study of algebraic curves and surfaces is very classical. Included
among the principal investigators are Riemann, Picard, Lefschetz, Enriques,
Severi, and Zariski. Beginning in the late 1940s, the study of abstract (not
necessarily algebraic) complex manifolds began to interest many mathe-
maticians. The restricted class of Kahler manifolds called Hodge manifolds
turned out to be algebraic. The proof of this fact is sometimes called the
Kodaira embedding theorem, and its proof relies on the use of the vanishing
theorems for certain cohomology groups on Kahler manifolds with positive
lines fundles proved somewhat earlier by Kodaira. This theorem is analogous
to the theorem of Riemann that a compact Riemann surface is algebraic.
This book is a revision and organization of a set of notes taken from the
lectures of Kodaira at Stanford University in 1965-1966. One of the main
points was to give the original proof of the Kodaira embedding theorem.
There is a generalization of this theorem by Grauert. Its proof is not included
here.
Beginning in the mid-1950s Kodaira and Spencer began the study of
deformations of complex manifolds. A great deal of this book is devoted to
the study of deformations. Included are the semicontinuity theorems and the
local completeness theorem of Kuranishi. There has also been a great deal
accomplished on the classification of complex surfaces (complex dimension
2). That material is not included here.
The outline is roughly as follows. Chapter I includes some of the basic
ideas such as surgery, quadric transformations, infinitesimal deformations,
deformations. In Chapter 2, sheaf cohomology is defined and some of the
completeness theorems are proved by power series methods. The de Rham
and Dolbeault theorems are also proved. In Chapter 3 Kahler manifolds
are studied and the vanishing and embedding theorems are proved. In Chapter
4 the theory of elliptic partial differential equations is used to study the
semi-continuity theorems and Kuranishi's theorem.
It will help the reader if he knows some algebraic topology. Some results
from elliptic partial differential equations are used for which complete
references are given. The sheaf theory is self-contained.
We wish to thank the publisher for patience shown to the authors and
Nancy Monroe for her excellent typing.
Seattle, Washington
January 1971
v
James A. Morrow
Kunihiko Kodaira

Contents
Preface v
Chapter 1. Definitions and Examples of Complex Manifolds 1
1. Holomorphic Functions 1
2. Complex Manifolds and Pseudogroup Structures 7
3. Some Examples of Construction (or Description) of
Compact Complex Manifolds 11
4. Analytic Families; Deformations 18
Chapter 2. Sheaves and Cohomology 27
1. Germs of Functions 27
2. Cohomology Groups 30
3. Infinitesimal Deformations 35
4. Exact Sequences 56
5. Vector Bundles 62
6. A Theorem of Dolbeault (A fine resolution of (I)) 73
Chapter 3. Geometry of Complex Maoifolds 83
1. Hermitian Metrics; Kahler Structures 83
2. Norms and Dual Forms 92
3. Norms for Holomorphic Vector Bundles 100
4. Applications of Results on Elliptic Operators 102
5. Covariant Differentiation on Kahler Manifolds 106
6. Curvatures on Kahler Manifolds 116
7. Vanishing Theorems 125
8. Hodge Manifolds 134
Chapter 4. Applications of Elliptic Partial Differential Equations to
Deformations 147
1. Infinitesimal Deformations 147
2. An Existence Theorem for Deformations I.
(No Obstructions) 155
3. An Existence Theorem for Deformations II. (Kuranishi's
Theorem) 165
4. Stability Theorem 173
Bibliography 186
Index 189
Errata 193
vii

[1]
Definitions and Examples
of Complex Manifolds
I. Holomorphic Functions
The facts of this section must be well known to the reader. We review
them briefly.
DEFINITION 1.1. A complex-valued function J(z) defined on a connected
open domain W s;;; en is called hoiomorphic, if for each a = (a1> "', an) e W,
J(z) can be represented as a convergent power series
+00
L ek, ... kn(Z1 - a1)k, ... (zn - a,,)k"
k,~O.kn~O
in some neighborhood of a.
REMARK. Ifp(z) =LCk ... kn (Z1 - a1)k, •.• (z" - an)k" converges at z =w, then
p(z) converges for any z such that IZk - akl < IWk - akl for 1 :S k :S n.
Proof We may assume a = O. Then there is a constant C> 0 such that
for all coefficients Ck.... kn '
Ie W"l .•• wknl < C
k, ..·kn 1 , , _ .
Hence
Ie zk, ... zknl < C 2 '" 2
IZ Ik' IZ Ik"
k, ... kn 1 , , - •
W 1 W"
(1)
If Izdwil < 1 for 1 :S i:S n, (1) gives
LIe", "'knZ~' '" zktl :S C.n( 1 I)< +00.
1=1 Zi
1- -
Wi
Q.E.D.
1

2 DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
We have the following picture:
Figure I
n is the region {zllzil < Iwd i ~ n}.
For convenience, we let
P(a,r) = {zllz. - a.1 < r., v = 1, "', n}.
Sometimes we call Pea, r) a po/ydisc or po/ycylinder. A complex-valued func-
tion/(z) = /(x1 + iYI, ... , Xn + iYn), where i = J - 1 can be considered as a
function of 2n real variables. Then:
DEFINITION 1.2. A complex-valued function of n complex variables is con-
tinuous or differentiable if it is continuous or differentiable when considered
as a function of 2n real variables.
We have:
THEOREM 1.1. (Osgood) If fez) = /(Zl' "', Zn) is a continuous function
on a domain W £ en, and if/ is holomorphic with respect to each z" when
the other variables Zi are fixed, then/is holomorphic in W.
Proof Take any a E Wand choose r so that pea, r) ~ W. We use the
Cauchy integral theorem for the representation for ZE Pea, r)
f( . . . ) - _1 f f(w l , z2, ... , Zn) d
ZI, , Z" - • J, wI>
2Xl Iw,-lId=r, WI - Zl
f( ... )- 1 f. f(w l ,W2 ,Z3,···,z")d
WI> Z2' ,Z" - - . W2,
2x! Iwz-lIzl=rz W2 - Z2
and so on.

1. HOLOMORPHIC FUNCTIONS 3
Substituting we get
We are assuming
Iz. - a'l < 1.
w. - a.
Hence the series
1 1 [ 1 ] 1
w. - Z. = (w. - a.) + (a. - z.) = 1 - (Zy - ay/w. - aJ w. - a.
( 1 ) 00 (Z - a )k
= L v v
w. - a. k=O w. - a.
converges absolutely in P(a, r). Integrating term by term we get
00
J(z) = L ct ! ••• kn(zt - a1)k! ••• (zn - an)kn, (2)
n=O
where
Then
where M = sup{IJ(w)llw E P(a, r)}. It follows that the representation (2) for
J(z) is valid for Z E P(a, r) and hence the theorem is true.
We now introduce the Cauchy-Riemann equations. Let/(z) be a differen-
tiable function on domain n f; en.
DEFINITION 1.3. The operators a/azy , a/oz., 1 ~ v ~ n are defined by
af 1 (aJ . Of)
o~. = 2 ax. - I oY. '
af 1 (af . OJ)
oz. = 2 OXy + I Oy. '
where z. = Xy + iy. as usual.

Let f(z) = u(x, y) + h'(x, y). Then
of = ~ [au + i av + i(au +i av)]
az 2 ox ax ay oy
= ~ [OU _ ov + i(OV + aU)].
2 ox oy ox oy
So, af/oz = °if and only if ou/ax = oll/ay and or/ax = -ou/ay (the Cauchy-
Riemann equations).
REMARK. If of/oz = 0, then df/dx = of/oz, where df/dx = ou/ox + i(ov/ox).
The following calculation verifies this:
of = ~ [au + i ov _ i(OU + i Ov)]
OZ 2 ax ax ay ay
= ~ [OU + i OV + i (av _ i au)] .
2 ax ax ax ax
THEOREM 1.2. Let fez) be a (continuously) differentiable function on the
open set Q s;;; en. Thenf(z) is holomorphic if and only if of/oz. = 0, i :s v :S n.
Proof This follows easily from Osgood's theorem and the classical
fact for functions of one complex variable. We need another simple calcula-
tion. From now on differentiable will mean having continuous derivatives
of all orders (C"").
PROPOSITION 1.1. Suppose few) =f(w1, ... , wm) and 9..(Z) I:s A.:s mare
differentiable and such that the domain offcontains the range of (91' ... , 9..)
= 9. Then f[91(Z), .. " 9m(Z)] is differentiable and if w;.(z) =9;.(z),
of = f (Of ow;. +!L ow;.)
oz. ..= lOW). oz. ow). oz. '
(3)
(4)
Proof All statements follow trivially from the chain rule of calculus.
For punishment we calculate (3). Let 11'). =U). + iv). = 9.(z). Then

I. HOLOMORPHIC FUNCTIONS 5
Making the substitutions,
1
U A = 2(g A + 9A),
we get
oj[g(Z)] = f {OJ! (09A + 09A)
OZ, A= I oUA2 oz. oz.
oj (1)(09A 09A)}
+ OVA 2i OZ. - OZ.
f {I (OJ . Of) ogA
= A= I 2 OUA - I OVA oz.
I (oj . of) 09A}
+- -+1- -
2 OUA OVA OZ.'
which gives (2).
COROLLARY 1. If f(w) is holomorphic in wand if w = g(z) = [gl(z), "',
gm(z)] where each g;.(z) is holomorphic in z, thenf[g(z)] is holomorphic in z.
COROLLARY 2. The set ()n of all functions holomorphic on n forms a ring.
In order to study complex manifolds we must consider holomorphic
maps. Let U be a domain in en and letfbe a map from U into em,
f(Zl' '.', zn) = [ft(z), ... ,fm(z)].
DEFINITION 1.4. f is holomorphic if each f;. is holomorphic. The matrix
ojl ojm
OZI OZI
= (iz:);.=I.....m
ojl ojm v= 1, ...• n
OZn OZn
is called the Jacobian matrix. If m = n, the determinant, det(of;./ozv) is called
the Jacobian. Writing out the real and imaginary parts W;. = U;. + iv;. =f;.,
z. = x. + iy., we have 2n functions U;., V;. of 2n real variables x., y•. We
write briefly

REMARK. IfI is holomorphic, a(u, v)/a(x, y) = Idet(al.,jaz.)IZ ~ o.
Proof We write it out for n = 2 and leave the general case to the reader.
We use the Cauchy-Riemann equations and set a.A= aUA/aX. = aVA/ay.,
bVA = aVA/aX. = -au}../ay•. Then
au, av, oUz avz
all bll al2 bl2
ax, ax, ax, ax,
av, = -b'l all -bJ2 al 2
aUI aU2 avz
ay, aYI ay, aYI
a21 b21 a22 b22
-b21 a2' -b22 a22
We perform the following sequence of operations: Multiplycolumn 2 by i and
add it to column I ; do the same with columns 4 and 3. Then multiply row 1
by i and subtract it from row 2; do the same with rows 3 and 4. Making use
of the fact that B.A = aIA/aZ. = a.A+ ib.A, we get
gil gl2 * *
a(u, v) gZI g22 * * = Idet(g.A)12
- - =
0 0 gl2
o(x, y) gil
0 0 gz, gZ2
by interchanging columns 2 and 3 and rows 2 and 3. Q.E.D.
THEOREM 1.3. (Inverse Mapping Theorem) Let/: V -+ en be a holomor-
phic map. If det(oJ,./oz.)lz=.. :F- 0, then for a sufficiently small neighborhood N
of a,Jis a bijective map N -+I(N);J(N) is open and/-'I/(N) is holomorphic
on/(N).
Proof The remark gives o(u, L,)/a(X, y) :F- 0 at a. We then use the inverse
mapping theorem for differentiable (real variable) functions to conclude that
I(N) is open, I is bijective, and I-I is differentiable on I(N). Set qJ(w) =
/-I(W); then z" = cp,,[J(z)]. Computing,
o= a~1l = ±aCPIl a~A + a~" a~A
az. A=I aw}.az. awAaz.
But det(alA/az.) = det(a/A/az.) :F- O. So by linear algebra, aqJ,,/aWA = 0 and
qJ =/-1 is holomorphic. Q.E.D.

2. COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES 7
COROLLARY. (Implicit Mapping Theorem) Letf)., A. = I, ... , m be holo-
morphic on V ~ en. Let rank (fJf)./fJz.) = r at each point z of V and suppose
in fact that det(iJf;./fJzvhsr :# o. Iff;.(a) = 0 for AS; m for some a E V, then in
vsr
a small neighborhood of a, the simultaneous equations,
have unique holomorphic solutions
AS; r.
For more details in this section one may consult Dieudonne (1960).
2. Complex Manifolds and Pseudogroup Structures
We assume given a paracompact Hausdorff space X which will also
generally be assumed connected. We want to define what we mean by a com-
plex structure on X (or structure of a complex manifold) which will be an
obvious generalization of the concept of a Riemann surface. First we want
to assume X is locally homeomorphic to a piece of C".
DEFINITION 2.1. By a local complex coordinate on X we mean a topological
homeomorphism z:p -+ z(p) E C" ofa domain U ~ X. z(p) = [Zl(p), ... , z"(p)]
are the local coordinates of X.
DEFINITION 2.2. By a system of local complex analytic coordinates on X
we mean a collection {Zj}jEI (for some index set I) of local complex co-
ordinates Zj: Vj -+ C" such that:
(I) X=UUJ •
JEI
(2) The maps fjk: Zk(P) -+ Zj(p) are biholomorphic [that is, Zj 0 Zk- 1 =
fjk and r;,/ = Zk 0 zj I are holomorphic maps from Zk( Vj n Vk) onto
Zj(Vj n Vk)] for each pair of indices (j, k) with Vj n Vk :# ljJ.
DEfiNITION 2.3. Two systems {Zj}jd' {II').}).'A are equivalent if the maps
Zj(p) -+ w).(p) are biholomorphic when and where defined.
DEfiNITION 2.4. By a complex structure on X we mean an equivalence class
of systems of local complex (analytic) coordinates on X. Bya complex mani-
fold M we mean a paracompact Hausdorff space X together with a complex
structure defined on X.

EXAMPLE, Complex projective space lPn, This is constructed from
en+1 _ {O} by identifying (p '" q)p = (pO, pI, ... , pn) and q = (qO, ... , qn) if
and only if pA = cqA for some nonzero c E C, for 0 ~ A. ~ n. Then IPn= en+ I -
{O}/'" is a compact Hausdorff space and one can construct a system of com-
plex coordinates as follows: We let Vj = {p E IPnlpj ¥- O}. Then {Vj}jsn is an
, f rrM 0 V th ( O J - I j+ I n) h
opencovenngo 10. n j emapzj= Zj,"',Zj ,Zj ,,,·,zj,were
z/ = pA/pj gives a local coordinate on Vj ; in fact, Zj(V) = en. Then
fjk: Zk --+ Zj is given by zj = z:/zt for A=F k, z~ = I/z{. (One simply multi-
plies by pk/pj ,) Thus we see that {Vj , zJ is a complex analytic system defining
a complex structure on IPn.
Generalizing this procedure we introduce the idea of a pseudogroup
structure. All spaces will be Hausdorff in what follows.
DEFINITION 2.5. A local homeomorphismf between two spaces X and Y is a
homeomorphism of an open set V in X to an open setf(V) in Y. One has a
similar definition of local diffeomorphism. A local homeomorphism (diffeo-
morphism) of X is such a map with X = Y.
Let 9 be a domain of IRnor en. Letfand 9 be local diffeomorphisms of 9.
If open W £:; 9, fl W denotes f restricted to W which is the restriction off to
domain (f) n W. If W is some open set such that 9 is defined on Wand
W nf(V) ¥- 4l. then 9 of is defined onf-I[W nf(V)],
feU)
Figure 2
DEFINITION 2.6. A pseudogroup of transformations in 9 is a set r of local
diffeomorphisms of 8 such that
(I) fEf=:.I- I Er.
(2) fE r, 9 E r = go IE r where defined.
(3) fE r=/1 WE r for any open W£:; 8.
(4) The identity map id E r.
(5) (completeness) Let I be any local diffeomorphism of 9. If [} = u Vj
andll Vj E r for eachj, thenfE r.

2. COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES 9
DEFINITION 2.7. Let r (a pseudogroup on 9) and X (a paracompact Haus-
dorff space) be given. By a system of local r-coordinates we mean a set
{ZjLd of local topological homeomorphisms Zj of X into 9 such that
Zj a Z;;l E r whenever it is defined. {w;.} and {Zj} are equivalent (f-equivalent)
if W;. a zj' E r when defined. A r-structure on X is an equivalence class of
systems of local r-coordinates on X. A r-manifold is a paracompact Haus-
dorff space X together with a r-structure on X.
EXAMPLES
1. 9 = en, re = (all local biholomorphic maps of e").Thenarc-struc-
ture is a complex structure, and a re-manifold is a complex manifold.
2. 9 = ~", fd = (all local diffeomorphisms of ~n). Then a fd-structure
is a differentiable structure and a fd-manifold is a differentiable manifold.
3. Let r be the set of a local diffeomorphism / of ~2" satisfying the
following condition. The matrix (e;..) will be defined to be
0 -1
1 0
0
0 -1
0
0 0 -1
1 0
where the blocks (?-~) occur on the diagonal and the rest of the entries are
zeros. If x = (x', ... , x2n) E ~2n,f(x) = [!t(x), ... ./2ix)] then the derivatives
of/ should satisfy
A system satisfying Example I is called a Hamiltonian dynamical system,
and such an / is a canonical trans/ormation. In this case a f-structure is
called a canonical structure.
4. Let r = (local affine transformations of ~"). These transformations
have the form
n
/A(X) = La! xY
+ b
Y= ,
where the a~, b;' are constants and the matrix (a~) is nonsingular. In this case
a f -structure is called flat affine structure.
If pseudogroup structures f, and f 2 are such that f, c f 2' then every
system of local f, coordinates is a system of local f2 coordinates, and f,
equivalence implies r 2 equivalence. Hence, every f,-structure determines a

f 2-structure. By assumption f c fd for all f. So every f-structure on X
determines a differentiable structure on X and every f-manifold is a differen-
tiable structure on X and every f-manifold is a differentiable manifold. The
f-structure M is defined on the differentiable manifold X.
The problem of determining the f-structures on a given differentiable
manifold M for given f is one of the most important (and difficult) problems
in geometry. It is known, for example, that if X is a compact orientable
differentiable surface (real dimension 2), then the only complex structures on
X are those of the classical Riemann surfaces. In case X = S2 (as a differen-
tiable manifold), then X = pI complex analytically (this is a classical fact).
If the underlying differentiable manifold X is diffeomorphic to pn, then one
conjectures that X = pn complex analytically [see Hirzebruch and Kodaira
(1957)J, and Kodaira and Spencer (1958). If S211 is the sphere with its usual
differentiable structure, it can be shown [Borel and Serre (1953) and Wu
(1952)] that s2n for n =/; 1,3 has no complex structure
1
2n + I
[s2n = {(Xl' ••• , X2n+l) i~2 xf, (Xl'···' X 2n +I ) E 1R2n+I}J.
For S2 there is the usual complex structure. It has been recently proved by
A. Adler (1969) that S6 has no complex structure. As a final example, let M
be a compact surface and let f+ be the pseudogroup of all local affine
transformations,
v = 1,2
such that
We have:
THEOREM 2.1. [Benzecri (1959)] If a f+ -structure exists on M, then the
genus of M is I. If M is not a torus, then M cannot be covered by any system
{(x), X])} of local coordinates such that lax~/ax;;1 is constant on Uj n Uk
for each pair of indices (j, k).
The proof will not be given here.
We continue with the definitions. Let M be a complex manifold, Wan
open set in M, and {Zj} a coordinate system. Then a mapping I: W ~ Cl is
holomorphic (difJerentiable, and so on) ifI 0 zj I is holomorphic (d(fJerentiable,
and so on) for eachj where defined. Let N be another complex manifold with
coordinates {II";.} and I: W -. N. Then I is holomorphic (differentiable, and so
on) if lI"A 0 I 0 zj I is holomorphic where defined.
DEFINITION 2.8. A subset SsM of a complex manifold is a (complex)
analytic subvariety if, for each S E S, there are holomorphic functions IA(P)
defined on a neighborhood lJ 3 S, 1 :::;; A. :::;; r, such that S n U = {p I/ip) = 0,

3. CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS 11
I ~ A. ~ r}. Then fA = 0, I ~ A. ~ n, are the local equations defining S at s.
The subvariety S is called a submanifold if S is defined at each s E S by local
equationsf. = 0 such that
k [ iJf;.(P)] .. d d f
ran - - = r IS 10 epen ent 0 s.
azj(p)
Suppose det(afA/azj)1 ~A:Sr =F O. Then letting
Isv,;,
w7(p) = lip),
w;(p) = z;(p),
for A= I, "', r
for A= r + 1, ... , n,
we have a local coordinate li'i = (wJ, "', wi» such that S: wJ = 11'] = ...
= wj = 0 (is defined by). Let (;(p) = IV'/A(p) = zj+A(p) for PES n Vj • Then
S is a complex manifold with local coordinates gj}'
We want to introduce meromorphic functions on a complex manifold.
They should be those functions which are locally quotients of holomorphic
functions. More precisely:
DEFINITION 2.9. A meromorphic function f on M is a complex-valued func-
tion defined outside of some proper subvariety S of M (S =F M) and such that
given q EM, there is a neighborhood V of q in M and local holomorphic
functions g, h on V such thatf(p) = g(p)/h(p) for p E V - S.
EXAMPLES
l. Any holomorphic mapf:M ..... [pI1 = C U {co}, [S =f-I(oo)].
2. In C2,j(Zl> Z2) = ZI/ZZ or f(zl' Z2) = P(ZI' Z2)/Q(Zl' Z2)' where P and
Q are polynomials.
3. Some Examples of Construction (or Description)
of Compact Complex Manifolds
First we have submanifolds of known manifolds ([pi", [pili X [pi", and so on).
Let [pi" have homogeneous coordinates «(0' "', (,,). Let fi0, I ~ A~ m be
homogeneous polynomials and define M = {( IfiO = 0, I ~ A~ m}. Suchan
M is called a projective algebraic (or simply algebraic) variety. If the rank of
(afA/a(.>c is independent of (E M, then M is a complex manifold. These are
exactly the classical algebraic (projective algebraic) manifolds. In some cases
the equationsfA = 0 give some easily read information about M. For instance,
iff is homogeneous of degree d, then Md = {Clfm = O} is called a hyper-
surface in [pi" of order d. If at least one of (ofliJ().)«) =F 0, I ~ A. ~ n, for each
( E Md, then Md is nonsingular.

EXAMPLES
1. Md S;; 1FD2 a nonsingular plane curve of order d is a Riemann surface
of genus 9 = td(d - 3) + I.
2. A nonsingular Md £; 1FD3. Md is simply connected and the Euler num-
ber X(Md) = d(d2 - 4d + 6). [The formulas in Examples 1 and 2 can be ob-
tained from Hirzebruch (1962), p. 91, Equation (5). They are well-known
classical formulas. The simple connectivity is also well known and it follows
from the Lefschetz theorems on hyperplane sections-see Milnor (I963),
p. 41.]
3. Let M £; 1FD3 be defined by
M = {((.(2 - (0(3 = 0, (0(2 - (~= 0, (~- (1(3 = O}.
We claim that M is complex analytically homeomorphic to pl. One can easily
check that the map fJ: IFDI --.IFD3 defined by fJ(t) = (t5, t~tl' tot~, tn where
t = (to , t,) E IFDI, is a biholomorphic map of IFDI onto M.
We remark that in the cases of complex or differentiable structures, sub-
manifolds give many examples; but for general i-structures one does not
usually get sub i-structures.
Second we get quotient spaces.
DEFINITION 3.1. An analytic automorphism of M is a biholomorphic map
of M onto M. The set of all analytic automorphisms of M forms a group 9
with respect to composition. Let G £; 9 be a subgroup.
DEFINITION 3.2. G is called a properly discontinuous group of analytic auto-
morphisms of M if for any pair of compact subsets K" K2 £; M, the set
{g E G IgK, n K2 =t= <p} is finite.
DEFINITION 3.3. G has no fixed points if for all 9 E G, 9 =t= 1, 9 has no fixed
points.
THEOREM 3.1. If G is properly discontinuous and has no fixed point, then
the quotient space MIG is a complex manifold in an obvious natural manner.
Proof We shall assume that M is connected (or a countable union of
connected manifolds) and paracompact. Hence, M is u-compact (a countable
union of compact sets). Let MIG = {Gp PEM}, where Gp = {g(p) pEG}
are the orbits of p E M. As notation set MIG = M*, Gp = p*. We shall show
that given q E M we can choose a neighborhood V of q such that PI' P2 E V,
PI =t= P2 gives P; =t= p;. In fact, there is V 3 q such that gV (" V = <p for all
9 E G, 9 =t= 1. M is locally compact so let VI => V2 => V3 ... be a base of rela-

tively compact neighborhoods at q. Then Fm = {g IgVmn Vm =F cp} is a finite
subset of G and Fm;;2 Fm +S' ;;2 •••• If 3gmE Fm, gm =F 1 for all m, then since
each Fm is finite, n Fm 3 g, 9 =F 1. Therefore, gVm n Vm =F cp, for all m and
Vm ~ q, gives g(q) = q, contradicting the nonexistence of fixed points. Hence
we cover M with open sets Vj such that PI' P2 E Vj implies pi =F p~ and
thus, Vj ~ V; = {p* IPE Vj} is I - 1. We give V; the complex structure
that Vj has. That is, if Zj: P ~ Zj(p) is a local coordinate on Vj , then
zj: P* ~ zj(p*) = Zj(P) gives a local coordinate on M*. The system {zj}
then defines a complex structure on M* and the topology of M* is just the
quotient topology for the map M ~ M*. Q.E.D.
EXAMPLES
1. Complex tori. Let M = cn. Take 2n vectors {WI' .", w2n }, W k =
(Wkl' "., wkn) E Cn so that the Wj are linearly independent over lit Let
2n
G={glg:z~g(z)=z+ Lmkwk,mkEZ}.
k=1
Tn = en/G isa (complex)torus of complex dimension n. Let n = I and arrange
it so that WI = I, W z = w, where the imaginary part of W is positive. Then
T=CI/G.
Figure 3
exp 2,,/ 2 I
We have a map C - C*, z~ w = e "z where C* = {zlz =F O}.lfwe first
take g(z) = z + mlw + m2 and then exponentiate, we get e2Iti(z+mlwl. So
exp 21ti 0 9 = oeml • exp 21ti where oe = e2"iw and g(z) = z + mlw + m2' and
0< lexl < 1since Im(w) > O. Looking a little closer we see we have the diagram
C~C*
·1 n".' I~'
C-C
which commutes. Hence, if we let G* = {g* Ig*: w ~ exmw, me Z}, we see
T = C/G = C*/G*.

Figure 4
2. Hop! manifolds. Let W = eN - {O} and G = {gIn Im E oZ, g(Wl' ... ,
wN) = «Xl 11'1' ••• , (XN wN), where °< l(Xvl < I}. Then WIG is a compact com-
plex manifold since it is easy to see that G is properly discontinuous and has
no fixed points on W. It is also easy to see that WIG is diffeomorphic to
Sl x S2N-l.
3. Let M be the algebraic surface (complex dimension 2) defined:
M = {( Ia+ (i + (i + ,~ = o} £ p3.
Let
G = {gm 1m = 0, 1,2,3,4 where g«(o, ... , (3)
= (p(o, p2(1, p3(2' p4(3) and p = e21ti/5}.
Then 9 is a biholomorphic map p3 ~ p3 and g5 = 1. Consider the fixed
points of gm on p3. They satisfy (0 = v ~ 3), (p,"(v+l) - c) (. = °and the
fixed points are (l, 0,0,0), (0, I, 0, 0), (0,0, I, 0), and (0, 0, 0, I). These
points are not on M so there are no fixed points on M and M/G is a complex
mamfold. We saw before that M is simply connected and X(M) = d(d2 -
4d +6) where d = 5. Therefore, the Euler number of M is 55. Then the fun-
damental group 1C1(M/G) ~ G and x(MIG) = II.
4. Last we have the classical examples of Riemann surfaces and their
universal covering surfaces. If S is a compact Riemann surface of genus 9 ~ 2,
the universal covering surface of S is the unit disk D = {z E e11lzl < I}. Then
S = D/G where each element of G is an automorphism of D and hence of
the form
. z - (X
g(z) = el8 - - ,
(Xz - I
I(XI < l.

Finally we consider surgeries. Given a complex manifold M and a com-
pact submanifold (subvariety) ScM, suppose we also have a neighborhood
W => S and manifolds S* c W* with W* a neighborhood of S*. Suppose
j: W* - S* -.... W - S is a biholomorphic map onto W - S. Then we can re-
place W by W* and obtain a new manifold M* = (M - W) u W*. More
precisely, M* = (M - S) u W* where each point z* e W* - S* is identified
with z = j(z*).
- f
[-~J
Figure 5
EXAMPLE I. Hirzebruch (1951) Let M = 1Jl>' X 1Jl>'. In homogeneous
coordinates, 1Jl>' = {C/ ( = «(0' ~,)}; = {C u {(Xl in inhomogeneous coordin-
ates, (= (d(o e C u too}. M = 1Jl>' X 1Jl>' = {(z, 0 Iz e 1Jl>', (e 1Jl>1} contains
S = to} X IJl>I and W = D X 1Jl>' where D = {zllzl < e} is a neighborhood of
Sin M. Let W* = D X 1Jl>'* = {(z, (*) IzeD, (* e 1Jl>'*} and S* = to} x 1Jl>*.
Fix an integer m > 0 and define j: W* - S* -.... W - S as follows:
j(z, (*) -.... (z, 0 = [z,«(*/z'")] where 0 < Izl < B.
Then j is biholomorphic on W* - S* and let M! = (M - S) u W* where
(z, 0 = (z, (*) if (* = zln(, 0 < Izl < f:.
REMARK. M and M! are topologically different if m is odd.
Proof (for m = I). M = 1Jl>' X IJl>I is homeomorphic to S2 x S2. We
show that the homology intersection properties of M and Mi are distinct,
hence, proving that they are topologically different. A base for HiM, Z) is
given by {SI' S2} where SI = to} X 1Jl>1, S2 = IJl>I X to}. Hence, any 2-cycIe C
is homologous ("') to as, + bS2 , a, b e Z. The intersection multiplicity
I(C, C) = J(aS, + bS2, aSI + bS2) = a2[(SI' SI) + b2[(S2 ,S2) + 2abl(SI' S2).
Since St. S2 occur as fibres in IJl>I x 1Jl>1, [(SI, S,) = [(S2' S2) = o. Hence,
I(C, C) = 2ab =0 (mod 2). (1)

In M we have the following picture:
w w'
M
./""~
I--V
s, s.
Figure 6
where At" is the submanifold of M~ defined by C= c and C* = zc with the
coordinates explained before. Then At" is a 2-cycle and Ao '" Ac. Hence
/(Ao, Ao) = /(Ao, Ac) = 1. Since for any 2-cycle Z on M, /(Z, Z) == 0 (mod 2)
we see M::f: Mr.
REMARKS
1. M!::f: M:(m ::f: n) as complex manifolds.
2. M~m = M topologically.
3. M~m+1 = M~ topologically.
These facts are proved in Hirzebruch (1951).
EXAMPLE 2. (LogarithmicTransformation) LetM = T x Pl,T = C/G,
G = {mw + n Im, n E 7L, 1m w > O} where T is a torus of complex dimension 1.
For any CE C, we denote the class in C/G = T by [C]' We perform surgery on
M as follows: Let S = {O} x T, W = D x T where p1 = C u {<X)} and
OED = {z E Clizi < e}.
w
T
s
Figure 7
Then set W* = D x T = {z, [(*] IzED, [(*] E T} and S* = {OJ x T £ D x
T. Define/: W* - S*~ W - S as follows:
/: (z, [(*]) ~ {z, [(* + (l/2ni) log z]},
where 0 < Izi < e.

Then f is biholomorphic and we can form M* = (M - S) u W*, where
(z, [C]) =(z, [(*]) if [C] = [(* + (1/2ni) log z], 0 < Izl < 6.
REMARK. For the first Betti numbers bl we have b2(M) = bz(T) = 2, but
bl (M*) = I. In fact, M* is topologically homeomorphic to S3 x sl.
Proof H2(M, Z) ~ 7L ffi 7L is clear by the Kunneth theorem. To study
M*, first we notice that M - W = (l?1 - D) x T is homeomorphic to a x T
where a is a closed disk, and T is homeomorphic to SI x Sl. If ( = x + yw,
we can identify [C] with (x, y), where x + 1 is identified with x, y + 1 with y,
where x and yare real (E IR). Therefore W* = D X Sl X Sl, M - W = a x
Sl x Sl. Since we are only interested for the moment in the topological type
of M* we may as well assume that D is the unit disk and that the identification
in the definition of surgery takes place on the boundary of D = {ei9 10 $ () $
2n}. Then we identify (w, x*, y*) and (w, x, y) if x = x* + «(}/2n), y = y*.
Hence, M* = B X SI where B is a circle bundle over S2 ; and in fact, we easily
see by the transition function that this is the Hopf bundle S3 -+ S2. Hence
B = S3. This proves that M* = S3 X Sl; b1(M*) = 1 follows.
EXAMPLE 3. We mention also the classical quadric transformation
(blowing up, u-process). First we discuss the case where M has complex di-
mension 2. Let S = p be any point on M, and let S* = pI be a copy of the
Riemann sphere. We define M* = (M - p) u pi as follows: Choose a co-
ordinate patch W = {(ZI' z2)llz11 < 6, IZ21 < 6} in a neighborhood ofp so that
Zl(P) =zz{p) =o. We define a submanifold W* of W x pI as follows:
W* = {(ZI' Z2; (I' (2) E W x pi IZI(2 - Z2 (I = O},
where ((1(2) are homogeneous coordinates on pl. W* is a submanifold since
(aflaz1) = (2' (af/azz) = -(I iff= ZtC2 - ZZ(l' and hence [(af/az1), (af/azz)]
:F (0, 0). Letf*: W* -+ W be the restriction of the projection map W x pi -+
W to W*. Then W* 20 X pi = S*, f*: S* -+p = (0,0), andf*: W* - S*-+
W - pis biholomorphic. The first two statements are obvious. For the proof
of the last, let (ZI' Zz; (10 (z) f/ S*. Then at least one of Zi:f: 0 and hence
«(I' (z) is determined by (Zl' ZZ)f*-I: (Zlo Z2) -+ (Zl' Zz; Zl, zz). By surgery we
obtain M* = (M - p) u pl. We make the following definition:
DEFINITION 3.4. The quadric transformation Qp with center p is the mani-
fold Qp(M) = M*.
REMARK. QPm··· Qp,(pZ) can be complicated! For example,
Qp6 ... Qp,(pz) = {( I(~ + ci + ,~ + (~ = O} S; p3.

For manifolds M of dimension ~ 2 we proceed analogously. If
dime M = n, let (Zl•... , zn)
be coordinates centered at pEp = (0, ···,0)]. If W = {(Zl' ... , zn)llz,,1 <
E, I $ C( $ n}, we set W* = {(z, c) IZ;. C. - Z. C
... = O,i $ A, v $ n} s;;; W x pn-l.
Again W* is a manifold, projection onto W defines a biholomorphic map
W* - pn-l -+ W - p, by (z, 0 -+ z. We form M* = (M - p) u W* =
(M - p) U pn-l and call M* = QP(M) the quadric transform of M with
center p.
4. Analytic Families; Deformations
Consider a torus Tro = CfG, G = {mw + n Im, n E 71. 1m w -+ O}. We have
a family of tori depending on the parameter w. Many examples of compact
complex manifolds depend on parameters built into their definitions. We also
have the examples of hypersurfaces of degree d in pn. Each such surface
Md = {C 1/<0 = O} is defined by a function I of the form 1=Lka+'.'+kn=d
aka ••. kn'io ... C~n. In a sense to be made precise Md depends" analytically"
on the coefficients aka •.• kn off We make the following definition:
DEFINITION 4.1. Let B be a (connected) complex manifold and let
{M,l t E B} be a set of compact complex manifolds depending on t E B. We say
that M, depends holomorphically (or complex analytically) on t and that
{M,l t E B} forms a complex analyticlamily if there is a complex manifold .It
and a holomorphic map (jj onto B such that
(I) (jj-l(t) = M, for each t E B, and
(2) the rank of the Jacobian of (jj is equal to the complex dimension of
B at each point of .It.
We note that (2) implies M, is a complex submanifold of .It.
Now for some examples. As before, we denote Tw = CfG,
G = {n + mw In, m E 71., 1m w > OJ.
Let B = {w 11m w > O} c: C. Let f§ = {9mn 19mn: (w, z) -+ (w, Z + mw + n)}.
Then f§ is a properly discontinuous group of transformations on B x C with-
out fixed point. Hence, .It = B x C/f§ is a complex manifold. The projection
map B x C -+ B induces a hoiomorphic map .It ..!! B, and (jj-I(W) = Tw. It is
easy to see that the Jacobian condition is satisfied so {T", IwEB} forms a
complex analytic family.
But suppose we proceed as follows: Again Tw =CfG and the map
C -+ CfG is written Z-+ [z]. Let D = unit disk = {tlltl < I}. On D x Tro con-
sider the group f§ = {I, 9} where 9: (t, [z]) -+ (- t, [z + !]) is of order 2.

4. ANALYTIC FAMILIES; DEFORMATIONS 19
Then I'§ is properly discontinuous and has no fixed points so D x T(J)/I'§ is a
complex manifold. Let 1t: D x T(J) -+ D be defined by (t, [z]) -+'t = t2• Then
the diagram
(t, [z]) ~ (-t, [z + t])
.j " j.
t2 _ _ t2
commutes so 1t defines a holomorphic map on .;It. The Jacobian condition
is not satisfied by 1t, since (j'J't/ot) = 2t = 0 at t = o. We notice that 1t- 1('t) =
T(J) if't =1= 0, but 1t -1(0) = T*, a torus of period w/2.
DEFINITION 4.2. Let M, N be compact complex manifolds. M is a deforma-
tion of N if there is a complex analytic family such that M, N s;;; {Mt It E B},
that is, M ta = M, Mtl = N.
We have the following sequence of problems to guide our work:
PROBLEM. Determine all complex structures on a given X.
PROBLEM. Determine all deformations of a given compact manifold
M.
PROBLEM. (easier?) Determine all "sufficiently small" deformations
of a given M.
DEFINITION 4.3. We say that all sufficiently small deformations have a cer-
tain property f!jJ if, for any complex analytic family {Mt It E B} such that
M ta = M, we can find a neighborhood N, to ENe B such that M t has f!jJ for
each tEN.
By standard techniques in differential topology we prove the following
theorem:
THEOREM 4.1. Let M t be a complex analytic family of complex manifolds
M t • Then M t and M to are diffeomorphic for any t, to E B.
Proof The reader will notice that we really only use the differentiability
of the map 1t: .;It -+ B, analyticity is not needed. In fact, we prove: Let .;It be
a differentiable family of compact differentiable manifolds such that the dif-
ferentiable map 1t: .;It -+ B has maximal rank (.;It and B are differentiable
manifolds). Then M t is diffeomorphic to M ta •

First we construct a Coo vector field 0 on a neighborhood of M,o in ..it
such that 1t induces 7t*(0) = a/as, where s is a member of a coordinate system
(s, x 2 , ••• ,xm) in a neighborhood of the point 10 e B chosen as follows:
Figure 8
We connect 10 and t by an embedded arcy: (-e,1 + e) -+ {yes) Is e(- e,1 +e)}.
A compactness argument shows that we can assume that I and 10 lie in the
same coordinate patch and since y is an embedding we can find a chart with
coordinate (s, 12 , ••. ,1m) around lo(to = (0, ... , 0), I = (s, 0, ... , 0». Because
of the rank condition, 1t-1(y) = 7t- I {(s, 0, ···,0) I -e < s < 1 + e}, is a
submanifold of ..it, and we can assume that (s, xf, ... , xj) are coordinates
of ..it for a given point of 7t-I (y) in some neighborhood qJJ of the point. Then
the vector field (a/as)j on qJj satisfies 1t.(a/as)j = a/as. Then if {Pj} is a parti-
tion of unity subordinate to {OIlj} (uOUj is a neighborhood of M,o)' the vector
field 0 = LJ pj(a/as)j satisfies our requirements.
For the second part of the proof we seek a solution of the differential
equation
d
ds xj(r) = 0j[x(r)], (1)
where 0j is the a-component of 0 in the coordinate patch qJj' with initial
conditions xj(O) = y", where (0, y2, ... , y") is some point close to (0, ···,0).
Ifs is small enough and Iyl is small enough, Equation (1) has a unique solution
xj(r, y) on some small interval. By compactness, we can assume that M,o c:
U jqJj' a finite union of such patches, and that in each qJj' (1) is satisfied for
Irl < jJ. where jJ. is independent ofj. If xj(r, y) is such a solution, let Xj = jjk(Xk)
and define ef[r,hiO, y)] uniquely on qJj ( qJk by
(2)
Then
dxj(r, y) = Laxj aef[r,jkj(O, y)]
dr /I axf or '

4. ANALYTIC FAMILIES; DEFORMATIONS 21
and by the uniqueness of the solution to (I)
x/I:, y) =jjk(Xk[T,f,.iO, y)]). (3)
Equation (3) implies that xC-t", y), ITI < p, y E M,O is a well-defined differen-
tiable map defined on M,O for each T, ItI < p., and x(O, y) = y. Let cpiy) =
x(t, y); then CPo = id (on M,o)' It is also easy to check that 1t[cpt(Y)] = yet)
since 1t.(0) = dlds. Hence, CPt maps M,O into My(t) (for small t). We can re-
peat this argument for My(t) and define t/lv: My(t) -+ My(t+v) and by uniqueness
get t/I-t 0 CPt = id, CPt 0 t/I_to = id. Since everything is differentiable, the
theorem is proved. Q.E.D.
REMARK. This argument is very old. For a treatment from the point of
view of Morse theory, see Milnor (1963). Sometimes this theorem is attributed
to Ehresmann (1947).
We consider some more examples of complex analytic families. The de-
pendence of the complex structure of MI on t E B can be complicated as we
shall see.
EXAMPLE I. Consider again the family of tori {Teo Iw E H} where
H = {w 11m w > O} and Teo = CjG, G = {mw + n Im, n E Z}. From the clas-
sical theory of Riemann surfaces we see that Teo and Teo' are conformally
equivalent if Wi = (aw + blew + d) where a, b, e, dE Z, and ad - be = l.
Let r§ be the group of transformations acting on H which have the form
aw + b
w-+ ,
ew + d
a, b, e, dE Z, ad - be = l.
Then it is easily seen that r§ is properly discontinuous on H. A fundamental
region IF for r§(ug§ = H, g§ n IF = cP if g ::f id) is given by the shaded
region in the figure below, hence Teo ::f Teo" if w ::f Wi and w, Wi E IF.
1
Figure 9

The elliptic modular function J defines a conformal map J: H/~ -+ C. So
Tw = Tw' if J(w) = J(w').
EXAMPLE 2. (n-dimensional tori) We give an outline of some of the
facts. A torus Tn = Cn/G where G = {L7~ 1 miwi Imi eZ} where WI'···' W211
are complex n-vectors linearly independent over R
(a) We can replace {wi} by any other linearly independent basis of G.
That is,
2n
wi). = L aikwU,
k=l
(4)
where aik e Z, det (aik) = 1 are also permissible generators of the lattice
(group) G.
(b) We may also introduce new coordinates in cn so Z). -+ 2)., where
n
2l = L ZvYvl, Yvl e C,
v=1
Then,
(5)
The resulting change from Equations (4) and (5) becomes
(6)
We may assume that W n+1, ••• , W2n are C-linearly independent. Hence by
some change of coordinates (Yv).), we can obtain
... w) (6)11
. • . W In (Yv).) = 6>nl
2nn
where I is the n x n identity matrix.
I
Q)ln)
Wnn , (7)
So we may assume (wij) = (~), where n = (wij) 1 ~ i, j ~ n and 1=
(b···~)·
(c) We can also break (aik) into pieces:
Then (4) takes the form
wi). = (ajk)(~) = (gD, n~ = An + B, n~ = cn + D.

4. ANALYTIC FAMILIES; DEFORMATIONS
If one assumes that 0; is invertible, then (~D(O;)-I = (~') where
0' = (AO + B)(CO + D)-I,
det(~ ~) = 1.
23
(8)
The following treatment will be a bit sketchy; for more details con-
sult Kodaira-Spencer II (1958). The fact that WI' "', W n , (l, 0, "', 0),
(0, I, 0, 0 ,), ... (0, ... , 0, I) are real linearly independent implies
det(~ ~) t= 0,
which is the same as (2it det [Im(wJ).)] t= o. Consider the space H = {O
det(Im 0) > O} [some sort of a generalization of 1m W > 0 in Example (1)].
Let C§ = the set of all transformations
0-+ (An + B)(CO + D)-I = n',
where (~ ~) E SL(n, I), the invertible integral matrices of determinant + I.
This group does not really act on H since it is possible for CO + D to be
singular; one should consult Kodaira-Spencer for more details. H should be
extended to something more general on which SL(n, I) acts. In any case,
Tn= T
n
-, if 0' = gO, 9 E C§.
We would like to form H/C§. But it turns out that C§ is not discontinuous. In
fact, for any open set U c H, there is a point n E U such that {gO Ig E 'Y} n U
is infinite. Hence, the topologial space H/C§ with the quotient topology is not
Hausdorff and hence certainly not even a topological manifold by the usual
definition.
We next give some examples of families {Mt 1t E B} such that M t = M
for t t= to and M to t= M.
EXAMPLE 3. A Hop! surface is a compact complex manifold ofcomplex
dimension two which has W = (:2 - {(O, O)} as universal covering surface.
More precisely, the Hopf surface M t is defined by Mt = WI Gt where Gt =
{gm ImEl} and g: (ZI' Z2) -+ (azl + tz2, aZ2)' that is, (::) -+ (~ :)(:~),
where 0 < lal < I and t E C. Then Mt is a compact complex manifold.
LEMMA 4.1. {Mt It E q is a complex analytic family.

Proof = {Mt It E C} = C x W/f, where f = {ym ImE Z}, and
y(D = (~ iDeJ Q.E.D.
We claim
(1) M, = Ml (complex analytically) for t #= O.
(2) Mo #= MI'
Proof of(1). We make the following change of coordinates:
Then the equation
implies that M1 = Mt when t #= O.
Proof of(2). First we prove a special case of Hartog's lemma.
LEMMA 4.2. Any holomorphic function defined on W = C2 - {CO, On can
be extended to a unique holomorphic function on C2 •
Proof Let f(zl' Z2) be the function on W. Pick a number r > 0, and
define the function
1 1. few, Z2)
F(Zl' Z2) = -. j dw,
2m Iwl=r w - ZI
for Izd < rand Z2 arbitrary. Then F(zl' Z2) is an analytic function in its cylin-
der of definition which is a neighborhood of (0, 0). If we can prove f = F
where both are defined, we will be finished. We know thatf(w, Z2) is holo-
morphic if Z2 #= O. So Cauchy's theorem gives
Fix Zl' 0 < Izd < r. Then F(ZI' Z2) = f(zl' Z2) for Z2 #= O. Both are analytic
in Z2; therefore,
F(zl' 0) = f(zi' 0).
Hence they agree where defined, proving the lemma.
Now let us suppose M, = Mo. I oF O. Then there is a biholomorphic map
f: M, -+ Mo. W is the universal covering manifold of M, and Mo. sofinduces

4. ANALYTIC FAMILIES; DEFORMATIONS
a mapI: W --t W which is biholomorphic, such that
W~W
G'l f IGo
Mt---+Mo
commutes.
It follows that Gt = 1-' Go! Hence for generator 9t of Gt ,
9, =1-'9"5' f.
Write the mapI in coordinates as
I(z" zz) = [J,(z" zz)'/z(z" Z2)].
25
(9)
Then by Hartog's lemma extend liz" zz) to a holomorphic function F;.
(z" zz) on CZ• Then F maps CZ into CZ [F =(F1, F2)], and F(O) =O. For if
not, extend 1-1 to F which satisfies F[F(z)] = z on Wand by continuity,
F[F(O)] = O. But if F(O) =F 0, £[F(O)] = I-I [F(O)] =F O. This contradiction
gives the result. Now expand F)"
F;,(ZI' zz) = F)"z, + F).,zz + F;'3ZT + F).• ZIZ2 + ....
We know thatf[9,(z)] =9"5 I [f(z)] so
Rewriting this gives
( O)±I
F[g,(z)J = ~ rx F(z).
F,(rxz, + lZ2 , rxZ2) = rx±IF,(ZI' zz),
Fz«(1.z1 + lz2 , cxzz) = rx±IP1(ZI, zz)·
Expanding these and taking the linear terms yields
(P
Il P'z) (rx t) = (rx 0)± 1 (Pll
Pli P1.2 0 rx 0 rx Fl ,
This can only happen when t = O. Hence M1 =F Mo. Q.E.D.
EXAMPLE 4. Ruled Surfaces (examples of surgery) Our ruled surfaces
will be IFDI bundles over IFD'. Let IFDI = {' I' E C U {oo}} (nonhomogeneous
coordinates). M(m) = VI x 1FD1 U Vl x.1FD1 where VI u Vl = IFDI, VI = C,
Vl = 1FD1 - {O}, and identification takes place as follows (recall Section 3):
Let (ZI' (I) E VI x 1FD1, (Zl' ~2) E Vl X IFDI. Then
REMARK. MC",) =F M(I) for m =F t' (not to be proved now).

THEOREM 4.2. M(t) is a deformation of M(m) if m - t =0 (mod 2). Assume
that In> f. Then there is a complex analytic family {M t II E C} such that
Mo = M(m) and M, = M(f) for 1:1= o.
Proof Define M, as follows: M, = VI X pI U V 2 X pI where (ZI' (I)
+-4(Z2' (2) if ZI = l/z2 , (I = Z~(2 + tz~ where k = !(m - t). Then it is easy to
see that {M, t E q is a complex analytic family and that Mo = M(m).
Suppose 1:1= O. Introduce new coordinates on the first PI by
(' _z~'I - t (linear fractional transformation).
1- 1(1
On the second pi,
r' '2
'>2 = I '" kv + t2·
22 C,2
Then, using ZIZ2 = I, and (I = Z~'(2 + IzL we get
Hence, in the new coordinates, ZI Z2 = I, (~ = z£(~; so
for t:f= O. Q.E.D.
PROBLEM. Finda pair ofcomplex analytic families {M,lltl < I}, {N,II/I < I}
such that
(a) Mo =1= No,
(b) M, = No for t =1= 0,
(c) N, = Mo for t =1= O.
(not complex analytically
homeomorphic)
There are no known examples of this type.

[2]
Sheaves and Cohomology
I. Germs of Functions
Let M be a complex (or differentiable) manifold. A local holomorphic
(differentiable)function isaholomorphic (differentiable) function defined on an
open subset U £; M. We write D<f) for the domain off Let p E M and suppose
given local functions f, g such that D(f) 11 D(g) 3 p. We say that rand g are
equivalent at p ifI(z) = g(z) for z E W £; D(!) 11 D(g), Wa neighborhood of
p. By a germ 01alunction at p we mean an equivalence class of local functions
at p. Denote by Ip the germ of1at p, (!)p the set of germs of all hoiomorphic
functions at p, and £1)p the set of germs of all differentiable functions at p.
The definitions
rxlp+ pgp = (af + pg)p
fp' gp = (fg)p,
rx, p E C,
are well defined, hence, (!)p, £1)p become linear spaces over IC. We also define,
We put a topology on (!) and El) as follows: Take any cp E (!) (or El); then
cp E (!)p (or El)p) for some p. Take any holomorphic (differentiable) 1 with
Ip = cp and define a neighborhood of cp as follows:
where p E U £; M, U is an open set in D(f). It is easy to see that the system of
neighborhoods (il1(cp;f, U) defines a topology on (!) (or £1).
EXAMPLE. (!) on the complex plane C. Let p E IC. Then if1 and g are
holmorphic at p we have expansions valid in some neighborhood of p,
co 00
fez) = L fk(z - pt, g(z) = Lgk(Z - pt,
k=O k=O
so1and g are equivalent at p if and only ifIk = gk for all k. Hence, the germ at
p is represented by a convergent power series; (!)p = ring of convergent power
series. And an element cp E (!)p can be represented by cp =Ip = {p;/o ,fl' ...}
where Iimk....oo I/kll/k < +00 and the radius of convergence is r(cp) = 1/ lim.
27

28 SHEAYES AND COHOMOLOGY
We define
0fI{(J); E) = {t/I It/I =I q , Iq - pi < E where 0 < E < r{(J)}.
In terms of our representation we calculate
00 00
I(z) = LIk(Z - p)k = I fm{Z _ q + q _ p)m
k;O 111;0
Hence
0fI«(J); E) = {t/I It/I = (q; go,"', gk" ..), I(q - p)1 < E
gk = m~k(;)fm(q - p)m-k}.
We note that t/I E d//{(J); E) means that t/I is a direct analytic continuation of (J).
The case of ~ on IR is not so simple. If (J) =Ip where I is a Coo function
atp,
III
j(x) = I fk(X - p)k +O(x _ p)m.
k=O
But I is not determined by the Ik'S since there exist COO functions I which are
not identically zero, but which have all derivatives zero at some point.
Define w: (!) (or ~) -. M by w«(!)p) = p.
PROPOSITION 1.1. (1) wis a local homeomorphism (that is, there exists
0fI such that w: 0fI«(J);f, U) -+ U is a homeomorphism).
(2) w-l(p) = (!)p (or ~p) (obvious).
(3) The module operations on w-l(p) are continuous (that is, IX(J) + IN
depends continuously on (J), t/I).
Proof (1) 0fI«(J);f, U) = {fq Iq E U} and w: /q -+ q is certainly 1 - 1. It
is obvious that wis continuous. To show that w- l is continuous, let OfI(w; g, V)
be a neighborhood of t/I = f . We want to find a neighborhood W of q so
4
that fw=W-I(W)EOfI(t/I; g,V) for wE W. We know that 9q =t/I =Iq , so /
and g are equivalent at q. Hence, 1= 9 in some neighborhood N of q. Let
W = N n V. Then/w =gw on W, so Iw E 0fI(t/I; g, V) for wE W. This proves
that the w- l is continuous.
(3) Let (J)=/p,t/I=gp' Then 1X(J)+pt/l={~f+pg)p. Let OfI{IX(J)+pt/l;
h, U) be a neighborhood of IX(J) + Pt/l. Then IX(J) + pt/l = hp = (IX! +pg)p so

1. GERMS OF FUNCTIONS 29
h =rxf+ flg in some neighborhood V£ U of p. Then if U E OJI(lp;J, V),
• E all(1/1; g, V), we have
rxu + fl. = rxfq + flgq
= (rxf + flg)q
= hq E OJI(rxlp + fll/l; h, V).
Since OJI(rxlp + fll/l; h, V) £ O//(rxlp +fll/l; h, U) we are done. Q.E.D.
We now give a formal definition. Let X be a paracompact Hausdorff
space.
DEFINITION 1.1. A sheaf Y over X is a topological space with a map w:
Y --. X onto X such that
(1) iii is a local homeomorphism [that is, each point s E 9' has a neigh.
borhood all such that w: OJI -+ w(OJI) c X is a homeomorphism onto an open
neighborhood of w(s)].
(2) iii-1(x), x E X is an R-module where R = 71., IR, C, or principal
ideal ring.
(3) The module operation (s, t) -+ rxs + flt is continuous on w-1(x)
where rx, fl E R.
(The reader can easily generalize this definition, but for our purposes it
suffices.) The set Y" = w-1(x) is called the stalk of Y over x.
EXAMPLES. (of sheaves)
(l) (!J on a complex manifold.
(2) ~ on a differentiable manifold.
(3) The sheaf over X of germs of continuous (Ill or C valued) functions.
(4) The sheaf over X of germs of constant functions.
In Example (4) 9' = X x C with the following topology: Let s = (x, z);
then OJI(s) = {(y, z) lyE U, z fixed}. If r -+ f(r) is a continuous map into Y of
1= {r Ia < r < b}, then f(l) = {(y, z) Iz fixed and y = w(f(r»r E l}. In other
words we give X x C the product topology where X has its given topology
and C has the discrete topology.
DEFINITION 1.2. Let U be a subset (usually open) of X. By a section u
of 9' over U we mean a continuous map x --. u(x) such that iii u(x) = x.
Suppose X = M, a complex (or differentiable) manifold; and suppose Y =
(!J (or ~). If f(z) is a holomorphic (or differentiable) function on U, then
u: p -+fp, p E U is a section.

30 SHEAVES AND COHOMOLOGY
PROPOSITION 1.2. Let (1: V ~ f/ be a section (f/ as above). Then (1 deter-
mines a holomorphic (or differentiable function) 1= I(z) on V such that
a(p) =Ip.
Proof a(p) E (!)p (or ~p). Hence there is a holomorphic (or differen-
tiable) g(z) defined on some neighborhood of p so that a(p) = gp. Since g
depends on p we write, g(z) = gCpl(z). Define1as follows:
I(p) = g(P)(p).
Then1is obviously well defined. Then
(1) I(p) is a holomorphic (differentiable) function on V.
Proof Take Wa neighborhood of p, W ~ V. Let dIJ = dlt[a(p); g(Pl, W]
= {(g(P Iq E W}. Since a is continuous, for any small neighborhood N
of p, N ~ W, we have a(N) ~ dIJ. Hence a(q) = (g(P . But we also
know a(q) = (g(q»q. Thus, (g(q = (g(P, and g(q)(z) = g(Pl(z) for z in a
small neighborhood V of q, V c N. But I(q) = g(q)(q) = g(p)(q) for q E V. So
I(z) = g(P)(z) for z E V and g(P) holomorphic (or differentiable) in V implies
that1is also.
(2) By definition a(p) = (g(Pl)p = Ip for each p E V. Q.E.D.
Hence we have the maps:
local holomorphic (differentiable) functions
t
germs
t
sections = holomorphic (or differentiable) functions.
reV, f/) will denote the R-module consisting of all sections of f/ over
V. We remark that reV, (!) are all holomorphic functions over V and
reV,~) are all differentiable functions over V. Let {V;.11 ~ A. ~ n} be a
finite family of open sets in X such that n V). ::f: ljJ. Let a). E rcV)., f/) and
IX;. E R. Then L IX). a). E reV, f/) where V = n V).. Let W be an open set and
a E rcV, f/) for some open set V. Then x -+ a(x), x E W n V defines a
section of rcW n V, f/). We denote this section by rwa and call it the
restriction of a to W n V.
2. Cohomology Groups
Let X be a Hausdorff paracompact space and let f/ be a sheaf over X.
Fix a locally finite covering Ii/i = {Vj } of X. A O-cochain CO on X is a set
CO ={aj} of sections aJ E reVj')' A 1-cochain C· = {ajd is a set of sections

2. COHOMOLOGY GROUPS 31
Ujk E qUj n Uk, f/) such that Ujk = -Ukj (skew-symmetric). A q-cochain
cq = {ujo '" A} is a set of sections ujo "'ik E qujo n ... n Uj.' f/) which
are skew-symmetric in the indices jo ... A. Let Cq(Olt) be the R-module of all
q-cochains. We define a map Cq(0lt)...!..Cq+1(0lt), the coboundary map as
follows: For O-cochains, t5Co = {Tjk} = {Uk -uJ where CO = {ud; for
l-cochains C1 = {ujd, t5C1 = {Tjk/} where Tjkl = Uu -ujl + ujk = Ujk + Ukl
+ Utj . In general, t5Cq = {Tjo ... j.+.} if Cq ={Ujo ... j.}, where
+ (_l)q+lujo "' j '
= L(-l)kujo ··· j~ ... i.+.' (1)
where 1 means "omit."
We denote the q-cocycles by
zq(Olt) = {CqIt5Cq =O}.
The q-cohomology group (with respect to Olt) is
(2)
We should remark that t5C is always skew-symmetric and t5t5 =0 so that
t5Cq-l(Olt) £; zq(Olt) and Equation (2) makes sense. The qth cohomology group
of X with coefficients in the sheaf f/ is defined to be
Hq(X, f/) = lim Hq(Olt, f/).
'II
This limiting process will now be explained. We say that the open covering
"Y = {V;J AeA of X is arefinementofOlt = {UJieJ if there isamap s: A -+ J such
that VA c U.(A) = Uj(A) , where we setj(A) = SeA). We define a homomorphism
where
It is easy to check that
n~ : (q(OU) -+ U("Y),
nt: {uio·"j.} -+ {TAO'" A.}'
t5n~ = n~ t5,
(3)
(4)
so that n~ maps zq(°lt) into zq("Y) and t5Cq- 1(fI) into t5Cq-l('f"). Hence n~
induces a homomorphian n~: Hq(:5II) --+ Hq('f").
LEMMA 2.1. n~: Hq(:Jlt) -+ W('f") is independent of the choice of map
s : A ...... J in the definition of refinement.

Proof First some notation: fix indices IXo, ••• , IXq E A. Let
A.
V = V; (')... (') V; VI = V; (')... (') V; (')... (') V;
Clo CEq , (10 fl.1 IZq ,
./"-... ./"-...
Uil = Uf(ao) (') ... (') Uf(aj) (') Ug(aj ) (') ••• U9(II/) (') ... (') Ug(aq) ,
and
Ui = Uf(II,) (') ... (') Uf(IIj) (') Ug(IIJ ) (') ••• (') Ug(II.) '
where!, g: A ~ J are two refining maps. Define a function (kU)A, ...A. by
q
(kuh, ... A. = p"f;o(-1)P- 1rVouf(A,)· .. f(Ap)g(Ap)·· ·g(A.) (5)
Let us call the maps n~, defined by f and g,f*, and g*. We claim that the
following equation holds:
[(ok + kO)]IIo ... a. = (g*u - 1*u)ao ... a•. (6)
The function ku is not necessarily skew-symmetric in its indices; so we
skew-symmetrize
-r~I ..·A =(k'-r)A' ... A =~, Lsgn(Al
• • q. P.I
Next we use (6) to see that
[(15k' + k'b)u]ao ... aq = (g*u - 1*u)IIo ... II•.
Hence, if bu = 0, bk'u = g*u - f*u E bCq- 1(1'). Hence,f* and g* induce the
same map, Hq(CJlt) ~ Hq(1'). Therefore we prove (6). The reader can easily
check the following calculations:
q
(bku)ao ... II = L(-1)'ry(ku)w··;, ... a.
• 1=0
= t(-1try [tt-l)iryt uf(IIo)··· f(IIi9(IIi) ... g{;;) ... g(a.)
{=o i=O
+ t (-l)i- 1r v{ uf(IIo) ... ;(a';)f(IIJ)g(IIJ) ... g(a.)]
i=t+ 1
(bku)aO···II. = ~ (-1)(+iryu/(IIo)···f(IIj)g(aJ) ... ~) ... g(a.)
)<t
'( 1){+i+ 1
+ L... - rv u/(.0) ... f(II,) ... f(IIj)g(II ) ... g(II ).
j>t J q
Similarly,
(kbu)IIo···II = I,(-1)j+tryuJ(IIo)···;(a';)···f(II·)9(IIJ) ··g(II)
q tSj J q
(7)
(8)

2.
Equations (7) and (8) give
COHOMOLOGY GROUPS
q
- Lry 0j(IIo) "'/(IIj)g(IIj) ... g(IIq)
j=O
33
= ry O"g(IIO) ... g(IIq) - rV 0"/(110) "'/(IIq) ' (9)
proving Equation (6). Q.E.D.
Knowing that the map n~ depends only on 1111 and "Y, we proceed to the
definition of the limit. We write 1111 < -H' if -H' is a locally-finite refinement of 1111.
Then < is a partial order and given 1111, "Y there is -H' so that 1111 < -H' and
"Y < -H'. Hence the set of all locally finite coverings of X forms a directed set
with respect to <, and the following equations can be verified (using Lemma
2.1):
n:= id,
n:,. = n~ 0 n~,
DEFINITION 2.1. Hq(X,!/) = lim Hq(l1I1, !/).
'"
REMARK. We recall the definition of the limit lim. We say that g, hE Hq
'"
(1111, !/) are equivalent if there exists -H' > 1111 such that n:,.9 = n:, h. Denote
the equivalence class of 9 by g. Let
Hq(I1I1,!/) = {g g E HQ(I1I1, !/)}.
The map 9 ~ 9 defines a homomorphism II"',
n~ : HQ(I1I1, !/) ~ HQ("Y, !/),
and n~ induces a homomorphism n~,
_.'" - -
11.,. : HQ(I1I1, !/) ~ HQ("Y, !/).
LEMMA 2.2. n~ is injective.
Proof n~g = 0 if and only if n:,. 0 n~ 9 = 0 for some W. So n:,. 9 =
oand 9 = O. Q.E.D.
Hence, identifying H4(11I1,!/) with n~H4(11I1, !/), we may consider
HQ(I1I1, !/) c H4("Y, !/) provided that 1111 < "Y. Then by definition,
H9(X, !/) = UHQ(I1I1, !/),
'"

and n'1': Hq(lJIt, f/) -t Hq(lJIt, f/) £;; Hq(X, f/) is a homomorphism of
Hq(lJIt, f/) into Hq(X, f/).
PROPOSITION 2.1. HO(X, f/) = rex, f/).
Proof By definition C- l = 0 so HO(IJIt, f/) = ZO(IJIt, f/).
ZO(IJIt, f/) = [0'10' = {O'j},O'j E r(Vj' f/), DO' = OJ.
But (j0' = 0 means O'j(z) - O'k(Z) = 0 on Vj n Vk. Hence O'(z) E r(X, f/), de-
fined by O'(z) = O'j(z) when ZE Vj , is meaningful. This proves HO(IJIt, f/) =
r(X, f/) and implies HO(X, f/) = r(X, f/). Q.E.D.
PROPOSITION 2.2. H"": HI(IJIt, f/) -t HI(X, f/) is injective.
COROLLARY. HI(X, f/) = UHI(IJIt, f/).
""
Proof (of the proposition). Suppose hE HI(IJIt, f/) = ZI(IJIt)/DCO(d/t).
Then h = {O')k}' O'jk E r( Vj n Vk, f/) where 0'ij + O'jk + O'ki = O. We want to
show that n""h = 0 implies h = O. n""h = 0 means Ii = 0 and this is true if and
only if n~h = 0 for some 1', l' > 1JIt. Let 1(/ = {WjA IWjA = Vi n V).}. Then
"If/" is a locally finite refinement of l' and n~h = n~ 0 n~h = O. Also 1(/ > d/t
since 1(/i). C Vi and we can use the maps(iA,) = ;in the definition of refinement.
Then we have
where
't(j).)(jll) = 'tjAjll = fW,.l.'" Wj,. O'ij'
Then n~v h = 0 implies {'t i).jll} = D{'t i).}' that is, 't i).jll = 'tjll - 't i).' Since
'tWIl = rW•.l.",W.,.O'ii = 0, we obtain 'till = 'ti). on Wi). n Will' Vi = U).WjA, and
't i = 't il' on Will defines an element 't i E r( Vi' f/). Then the equation 0'ij = 'tj -
't i implies h = O. Q.E.D.
Consequently, in order to describe an element of HI(X, f/), it is sufficient
to give an element of HI(IJIt, f/) for some 1JIt.
EXAMPLE. Let M = {(Zl' z2)llzll < 1, IZzl < 1, (Zl' Z2) =F (0, O)}. Then
dime Hl(M, l!J) = + 00.
Proof. Set
VI = {(ZI' zz) I(ZI' zz) E M, ZI =I:- OJ,
VZ = {(Zl' zz) I(Zl' Z2) E M, Z2 =I:- OJ.

3. INFINITESIMAL DEFORMAnONS 35
In this case M = UI U Uz so chose as covering 0/1 = {VI' Uz}. Then
HI(o/1, (9) = ZI(OlI, (9)/bCO(o/1, (9) where ZI(o/1, (9) = {0'121 0'12 E r(VI () Uz ,
(9)},Co(o/1, (9) = {t It = (tl' tz), tit E r(V"' (9)}, and bCo(o/1, (9) = {tz - tl Itit E
r(V", (O)}.
We note that VI () Uz = {(ZI, zz) 10< IZII < 1,0 < IZzl < I}, so we have
a Laurent expansion for 0'12
m=-CX)n=-co
tl IS holomorphic on VI = {(ZI' zz) 10 < IZII < I, IZzl < I} so tl(Z) =
L~~-ooL:'=obm"z/~z~. Similarly for tz, tz(z) = L~=oL:=OO_oocm"z~z~, and
tz - tl = Lm~oor"2:0 am"z~z~. Then HI (0/1) ~ {0'121 0'12 = L;;;! -00 L;:!- 00
am"z'~z~}. Hence dim HI (0/1, f/) = +00 and since HI(o/1, f/) £;; HI(X, f/),
dim HI(X, f/) = +00. Q.E.D.
PROPOSITION 2.3. If HI( Vj, f/) = 0 for all Vj E 0/1, then HI(o/1, f/) ~
HI(X, f/) where d/I = {VJ.
Proof. We already know that HI(o/1, f/) £;; Hl(X, f/). Hence we only
need to show the following. Let "Y = {VA} be any locally finite covering.
Let if" = {WjAI WjA = Vj () VA}' Then it suffices to show that n:" :HI (0/1) -+
Hl("/Y) is surjective. Take a I-cocycle {O'jAb} of HI("/Y) where O'jAjlt + O'j"kv +
O'k,jA = O. Then {O'WIt} for each fixed i is a I-cocycle on the covering {Wj)J of
Uj' Since HI(Vj, f/) = O,HI({WU},f/) £;; HI(V j , f/)givesHl({Wu}, f/) = 0
for each i.This implies the existence oftiA E r( Wu , f/) such that aWIt = tjlt -
tiA' Let t be the O-cochain {tiA} on "/Y. Then {aIAh} = {aUk,} - bt defines a
I-cocycle on "/Y which defines the same cohomology class in Hl("/Y) as 0'.
From the definition of t we see that 0'1J.i1t = O. So O'iAj" + O'iltkv + O'~,jA = 0
yields O'lltkv = alAh' Similarly, O'jAh = O';ltkw' Hence, O'ik = aiAkv = ai,kv' and
O'ik E r( V j () Uk, f/). Now we have found aik so that n:.(O'tk) = O'tU" and
{aIAkv} is cohomologous to {ajAkv}' Hence n::,. is surjective. Q.E.D.
3. Infinitesimal Deformations
Using cohomology groups we will give an answer to the following
problem: Let .;II = {M1ft E B} be a complex analytic family ofcompact com-
plex manifolds M I and let t = (tl, ... ,t") be a local coordinate on B. The
problem is to define (aMI/at').
For this we define the sheaf of germs of holomorphic vector fields. Let M
be a complex manifold and let W be an open subset of M. Let 0/1 = {Vj, Zj}

be a covering of M with coordinates patches with coordinates p --+ Zj(p) =
[zl(p), .. " z7(p)]. A holomorphic vector field () on W is given by a family of
holomorphic functions {OJ} on W (' Vj where
n a
0= L OJ(p)-IX
IX=I aZj
on W (' Uj • These functions should behave as follows: On W (' U",
n a
0= L Of(p)p'
(1= 1 az"
We want
so the transition equation
(1)
should be satisfied on W (' Uj (' U". Thus we have a definition of local
holomorphic vector fields and we can define germs of local holomorphic
vector fields. As notation we denote by 0 the sheaf over M of germs of holo-
morphic vector fields. (Later we shall give a formal definition of the holo-
morphic tangent bundle of a complex manifold.)
Next we want to define the infinitesimal deformation (aM,/at.). First we
consider the case B = {tlltl < r} £; C. .I{ is a complex manifold and iij:
.I{ --+ B is a holomorphic map satisfying the usual conditions
(1) M, = i.ij-I(t);
(2) the rank of the Jacobian of iij = 1 = dim B.
We can find an e > 0 small enough so that iij-I(A), A = {tlltl < e} looks as
follows:
J
iij-I(A) = UOUj
j= 1
(a union of a finite number
of open sets).
On each OUj there should be a coordinate system
p --+ [z}(p), .. " zj(p), t(p)],
where t(p) = iij(p) and such that OUj = {pi Izj(p)1 < ej. It(p)1 < e}. We write
p = (Zj' t) = (z}, ... , zj, t). This construction is possible because rank iij = 1
These charts are holomorphically related so
zj(p) = fj,,[z~(p), .", z~(p), t(p)] = fj"(z,, , t)
on Uj (' Uk' Let U'j = M, (' OUj , It I < e. Then set
{(z} "', zj, t)llzjl < ej } = V'j'

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CHAPTER XVIII.
EVIL TIDINGS.
To those who have not experienced the coming of sudden disaster,
word descriptions are feeble. It is easy to tell how this and that
occurred; to speak of the wails and cries of the injured; to try to
depict the scene in sturdy English, but the soul-thrilling terror, the
horror, and physical pain of the moment must be felt.
In the present case the accident was so entirely unexpected that the
very occurrence carried an added quota of dreadful dismay. The spot
had never been considered unsafe. At the time of construction
eminent engineers had decided that it would be perfectly feasible to
lay the rails close to the edge of the sea.
A stout parapet of stone afforded ample protection, in their opinion,
but they had not gauged the resistless power of old ocean. The
coming of a fierce south wind worked the mischief, and in much less
time than is required in the telling, the doomed train was cast a
mass of wreckage against the unyielding face of the cliff.
The first crash extinguished the lights, adding impenetrable darkness
to the scene. It found Nattie and Mori within touch of each other.
They instinctively grouped together; but a second and more violent
wrench of the coach sent them flying in different directions.
The instinct of life is strong in all. The drowning wretch's grasp at a
straw is only typical of what mortals will do to keep aglow the vital
spark.
Terror-stricken, and stunned from the force of the shock, Nattie still
fought desperately for existence. He felt the coach reeling beneath
his feet, he was tossed helplessly like a truss of hay from side to

side, and then almost at his elbow he heard a familiar voice
shrieking:
"Mercy! mercy! The blessed saints have mercy upon a poor sinner.
Oi'm sorry for me misdeeds. Oi regret that Oi was even now going
against the law. Oi confess that Oi meant to lead them two young
fellows away so that——"
The words ended in a dreadful groan as the car gave a violent lurch,
then Nattie felt a shock of pain and he lost consciousness. When he
came to, it was to find the bright sun shining in his face.
It was several moments before he could recognize his surroundings.
A sound as of persons moaning in agony brought back the dreadful
truth. He found himself lying upon a stretcher, and near at hand
were others, each bearing a similar burden.
The temporary beds were stretched along the face of the cliff. A
dozen feet away was a huge mass of shattered coaches and the
wreck of a locomotive. A number of Japanese were still working
amid the débris, evidently in search of more victims of the disaster.
Nattie attempted to rise, but the movement caused him excruciating
pain in the left shoulder. A native, evidently a surgeon, was passing
at the moment, and noticing the action, he said, with a smile of
encouragement:
"Just keep quiet, my lad. You are all right, merely a dislocation. Do
not worry, we will see that you are well taken care of."
"But my friend?" replied the boy, faintly. "His name is Mori Okuma,
and he was near me when the accident occurred. Can you tell me
anything of him? Is he safe?"
"Is he one of my countrymen, a youth like yourself, and clad in
tweed?"
"Yes, yes."

"Well, I can relieve your anxiety," was the cheering reply. "He is
working like a trooper over there among the coaches. It was he who
rescued you and brought you here. Wait; I will call him."
A moment later Mori made his appearance, but how sadly changed
was his usually neat appearance. His hat was gone, his clothing torn
and disordered, and his face grimed with dust and dirt. He laughed
cheerily, however, on seeing Nattie, and made haste to congratulate
him on his escape.
"This is brave," he exclaimed. "You will soon be all right, old boy. No,
don't try to get up; your arm is dislocated at the shoulder, and
perfect quiet is absolutely necessary."
"But I can't lie here like a stick, Mori," groaned the lad. "What's a
dislocation, anyway? It shouldn't keep a fellow upon his back."
"You had better take the doctor's advice. The relief train will start for
Kobe before long, and once in a good hotel, you can move about.
This is a terrible accident. Fully twenty persons have lost their lives,
and as many more wounded."
"Have you seen anything of Patrick Cronin?"
"No, nothing. It is thought several bodies were carried out to sea
when the water rolled back after tearing away the parapet. His may
be one of them."
The Irishman's words, heard during the height of the turmoil,
returned to Nattie. He now saw the significance of the Irishman's
cry.
"Something is up, Mori," he said, gravely, explaining the matter. "It
certainly seems as if Patrick was leading us on a wild-goose chase."
"That was Grant's impression, anyway. Did the fellow really use
those words?"

"Yes, and he evidently told the truth. He was in fear of death, and
he confessed aloud that he was leading us away so that something
could happen. At the interesting moment his voice died away to a
groan, then I lost consciousness."
"What do you think he could have meant?"
"It is something to do with the Blacks, I'll wager."
"But does he know them?"
"He is acquainted with Willis Round, and that is the same thing."
Mori seemed doubtful.
"You don't think he intended to lead us into a trap?" he asked,
incredulously.
"Hardly, but——"
"Grant?"
Nattie sat up in the stretcher despite the pain the effort caused him.
"Mori, we must communicate with him at once," he said. "There is
no telling what could happen while we are away. Confound it! I'll
never forgive myself if this should prove to be a ruse. Can you
telegraph from here?"
"No, we must wait until we reach Kobe. Now don't excite yourself,
my dear fellow. You will only work into a fever, and that will retard
your recovery. I really think we are mistaken. But even if it should
prove true, it won't mend matters by making yourself worse."
The lad fell back with a groan. He acknowledged the wisdom of
Mori's remark, and he remained quiet until the relief train finally
carried him with the balance of the survivors to the city they had
recently left. Mori hastened to the telegraph office after seeing his
charge to a hotel.

What Nattie suffered in spirit during the Japanese youth's absence
can only be measured by the great love he bore his crippled brother.
The very thought that something had happened to him was anguish.
He knew that Grant was bravery itself despite his physical disability,
and that he would not hesitate to confront his enemies single-
handed.
When the turning of the door knob proclaimed Mori's return, Nattie
actually bounded from the bed and met him halfway. One glance at
the Japanese youth's face was enough. Evil news was written there
with a vivid brush. In one hand he held a telegram, which he gave
to his companion without a word.

CHAPTER XIX.
BAD NEWS CONFIRMED.
Nattie took the telegram with a sinking heart. He had already read
disquieting news in Mori's face, and for a moment he fumbled at the
paper as if almost afraid to open it. Finally mustering up courage, he
scanned the following words:
"Message received. Grant cannot be found. He left office at
usual time last night, but did not appear at his home. Have
done nothing in the matter yet. Wire instructions. Sorry to hear
of accident."
It was signed by the chief bookkeeper, a Scotchman, named Burr. He
was a typical representative of his race, canny, hard-headed, and
thoroughly reliable. Sentiment had no place in his nature, but he
was as impregnable in honesty as the crags of his own country.
Poor Nattie read the telegram a second, then a third time. The
words seemed burned into his brain. There could be only one
meaning: Grant Manning had met with disaster. But where, and
how? And through whom? The last question was easily answered.
"Mori," he said, with a trembling voice, "this is the work of the
Blacks and that scoundrel, Willis Round."
"Something may have happened, but we are not yet certain,"
gravely replied the Japanese youth. "Surely Grant could take a day
off without our thinking the worse."
"You do not know my brother," answered the lad, steadfastly. "He
hasn't a bad habit in the world, and the sun is not more regular than
he. No, something has happened, and we must leave for Yokohama
by the first train."

"It is simply impossible for you to go," expostulated Mori. "The
doctor said you must not stir from bed for three days at the very
least. I will run down at once, but you must remain here."
"If the affair was reversed, Grant would break the bounds of his
tomb to come to me," Nattie replied, simply. "Send for a surgeon
and ask him to fix this shoulder for traveling. I want to leave within
an hour."
The young Japanese threw up both hands in despair, but he left
without further words. In due time the man of medicine appeared
and bandaged the dislocated member. A few moments later Nattie
and Mori boarded the train for the north.
As the string of coaches whirled through valley and dell, past paddy
fields with their queer network of ridges and irrigating ditches; past
groups of open-eyed natives dressed in the quaint blue costumes of
the lower classes; through small clusters of thatched bamboo
houses, each with its quota of cheerful, laughing babies, tumbling
about in the patches of gardens much as the babies of other climes
do, Nattie fell to thinking of the great misfortune which had
overtaken the firm.
"If something has happened to Grant—which may God forbid—it will
be greatly to the interest of Jesse Black," he said, turning to his
companion. "Everything points in their direction. The first question in
such a case is, who will it benefit?"
"You refer to the army contracts?"
"Yes. It means to the person securing them a profit of over one
hundred thousand dollars, and that is a prize valuable enough to
tempt a more scrupulous man than the English merchant."
"I think you are right. If Grant has been waylaid, or spirited away,
which is yet to be proven, we have something to work on. We will
know where to start the search."

Yokohama was reached by nightfall. Mori had telegraphed ahead,
and they found Mr. Burr, a tall, grave man with a sandy beard,
awaiting them. He expressed much sympathy for Nattie's condition,
and then led the way to the jinrikishas.
"I can explain matters better in the office," he said, in answer to an
eager question. "'Tis an uncou' night eenyway, and we'll do better
under shelter."
Compelled to restrain their impatience perforce, his companions sank
back in silence and watched the nimble feet of the karumayas as
they trotted along the streets on the way to the Bund.
Turning suddenly into the broad, well-lighted main street, they
overtook a man pacing moodily toward the bay. As they dashed
past, Nattie glanced at him; then, with an imprecation, the lad stood
up in his vehicle. A twinge of pain in the disabled shoulder sent him
back again.
Noting the action, Mori looked behind him, and just in time to see
the man slip into a convenient doorway. It was Mr. Black.
"Keep cool, Nattie," he called out. "Confronting him without proof
won't help us."
"But did you see how he acted when he caught sight of us?"
"Yes, and it meant guilt. He tried to dodge out of our sight."
On reaching the office, Mr. Burr led the way inside. Lighting the gas,
he placed chairs for his companions, and seated himself at his desk.
"Noo I will explain everything," he said, gravely. "But first tell me if
ye anticipate anything serious? Has Mr. Grant absented himself
before?"
"Never," Nattie replied to the last question.
"Weel, then, the situation is thus: Last night he left here at the usual
hour and took a 'rikisha in front of the door. I was looking through

the window at the time, and I saw him disappear around the corner
of Main Street. I opened the office this morning at eight by the
clock, and prepared several papers and checks for his signature.
Time passed and he did na' show oop.
"At eleven I sent a messenger to the house on the 'bluff.' The boy
returned with the information from the servants that Mr. Grant had
not been home. Somewhat alarmed, I sent coolies through the town
to all the places where he might have called, but without results. I
received your telegram and answered it at once. And that's all I
know."
The information was meager enough. Nattie and Mori exchanged
glances of apprehension. Their worst fears were realized. That some
disaster had happened to Grant was now evident. The former sprang
to his feet and started toward the door without a word.
"Where are you going?" asked the Japanese youth, hastily.
"To see Mr. Black," was the determined reply. "The villain is
responsible for this."
"But what proof can you present? Don't do anything rash, Nattie. We
must talk it over and consider the best plan to be followed. We must
search for a clew."
"And in the meantime they will kill him. Oh, Mori, I can't sit here and
parley words while my brother is in danger. I know Ralph Black and
his father. They would not hesitate at anything to make money. Even
human life would not stop them."
"That may be. Still, you surely can see that we must go slow in the
matter. Believe me, Grant's disappearance affects me even more
than if he was a near relative. I intend to enter heart and soul into
the search for him. Everything I possess, my fortune, all, is at his
disposal. But I must counsel patience."
The tears welled in Nattie's eyes. He tried to mutter his thanks, but
his emotion was too great. He extended his hand, and it was

grasped by the young native with fraternal will. The Scot had been
eying them with his habitual placidity. The opening of a crater under
the office floor would not have altered his calm demeanor.
"Weel, now," he said, slowly, "can you no explain matters to me? I
am groping about in the dark."
"You shall be told everything," replied Mori.
He speedily placed him in possession of all the facts. Mr. Burr
listened to the story without comment. At the conclusion he said, in
his quiet way:
"I am no great hand at detective work, but I can see as far thro' a
millstone as any mon with twa gude eyes. Mister Grant has been
kidnaped, and ye don't need to look farther than the Black's for a
clew."
"That is my opinion exactly," exclaimed Nattie.
"I am with you both," said Mori, "but I still insist that we go slow in
accusing them. It stands to reason that to make a demand now
would warn the conspirators—for such they are—that we suspect
them. We must work on the quiet."
"You are right, sir," agreed Mr. Burr.
"What is your plan?" asked Nattie, with natural impatience.
"It is to place Mr. Burr in charge of the business at once, and for us
to start forth in search of possible clews. I will try to put a man in
the Black residence, and another in his office. We must hire a
number of private detectives—I know a dozen—and set them to
work scouring the city. The station master, the keeper of every road,
the railway guards, all must be closely questioned. And in the
meantime, while I am posting Mr. Burr, you must go home and keep
as quiet as you can. Remember, excitement will produce
inflammation in that shoulder, and inflammation means many days in
bed."

The authoritative tone of the young Japanese had its effect.
Grumbling at his enforced idleness, Nattie left the office and
proceeded to the "bluff." Mori remained at the counting-room, and
carefully drilled the Scotchman in the business on hand.

CHAPTER XX.
THE MAN BEYOND THE HEDGE.
It was past midnight when he finally left with Mr. Burr, but the
intervening time had not been wasted. Orders, contracts and other
details for at least a week had been explained to the bookkeeper,
and he was given full powers to act as the firm's representative.
After a final word of caution, Mori parted with him at the door, and
took a 'rikisha for the Manning residence. He found Nattie pacing the
floor of the front veranda. The lad greeted him impatiently.
"Have you heard anything?" he asked.
"Not a word. I have been busy at the office since you left.
Everything is arranged. Mr. Burr has taken charge, and he will
conduct the business until this thing is settled. We are lucky to have
such a man in our employ."
"Yes, yes; Burr is an honest fellow. But what do you intend to do
now?"
"Still excited, I see," smiled Mori. He shook a warning finger at the
lad, and added, seriously: "Remember what I told you. If you
continue in this fashion I will call a doctor and have you taken to the
hospital."
"I can't help it," replied Nattie, piteously. "I just can't keep still while
Grant is in danger. You don't know how anxious I am. Let me do
something to keep my mind occupied."
"If you promise to go to bed for the rest of the night I will give you
ten minutes now to discuss our plans. Do you agree?"
"Yes; but you intend to remain here until morning?"

"No, I cannot spare the time. I must have the detectives searching
for clews before daylight."
"Mori, you are a friend indeed. Some day I will show you how much
I appreciate your kindness."
"Nonsense! You would do as much if not more if the case was
reversed. Now for the plans. To commence, we are absolutely
certain of one thing: Patrick Cronin was in the scheme, and he was
sent to get us out of the way while Ralph and Willis Round attended
to Grant."
"I am glad the Irishman met with his just deserts," exclaimed Nattie,
vindictively. "He is now food for fishes."
"Yes; a fitting fate. The accident cannot be considered an unmixed
catastrophe. If it had not occurred we would have gone on to
Nagasaki, and have lost much valuable time. As it is, we are
comparatively early. What we need now is a clew, and for that I
intend to begin a search at once."
"Would it do any good to notify the American Consul?"
"No; our best plan is to keep the affair as quiet as possible. We will
say nothing about it. If Grant is missed we can intimate that he has
gone away for a week.
"Now go to bed and sleep if you can," he added, preparing to leave.
"I will call shortly after breakfast and report progress."
With a friendly nod of his head he departed on his quest for
detectives. Nattie remained seated for a brief period, then he walked
over to a bell-pull, and summoned a servant. At his command the
man brought him a heavy cloak, and assisted him to don his shoes.
From a chest of drawers in an adjacent room the lad took a revolver.
After carefully examining the charges he thrust it into his pocket and
left the house.

The night was hot and sultry. Not a breath of wind stirred, and the
mellow rays of a full moon beamed down on ground and foliage,
which seemed to glow with the tropical heat. Notwithstanding the
discomfort Nattie drew his cloak about him and set out at a rapid
walk down the street leading past the Manning residence.
From out on the bay came the distant rattle of a steamer's winch.
The stillness was so oppressive that even the shrill notes of a
boatswain's whistle came to his ears. An owl hooted in a nearby
maple; the melancholy howl of a strolling dog sounded from below
where the native town was stretched out in irregular rows of
bamboo houses.
The lad kept to the shady side of the road, and continued without
stopping until he reached a mansion built in the English style, some
ten or eleven blocks from his house. The building stood in the center
of extensive grounds, and was separated from the street by an
ornamental iron fence and a well-cultivated hedge.
It was evidently the home of a man of wealth. In fact, it was the
domicile of Mr. Black and his son Ralph. What was Nattie's object in
leaving the Manning residence in face of Mori's warning? What was
his object in paying a visit to his enemy at such an hour of the
night?
Anxious, almost beside himself with worry, suffering severely from
his dislocated shoulder, and perhaps slightly under the influence of a
fever, the lad had yielded to his first impulse when alone, and set
out from home with no settled purpose.
On reaching the open air he thought of Jesse Black. The mansion
was only a short distance away; perhaps something could be learned
by watching it. The conjecture was father to the deed.
Selecting a spot shaded by a thick-foliaged tree, Nattie carefully
scanned the façade of the building. It was of two stories, and
prominent bow-windows jutted out from each floor. The lower part

was dark, but a dim light shone through the curtains of the last
window on the right.
A bell down in the Bund struck twice; it was two o'clock. At the
sound a dark figure appeared at the window and thrust the shade
aside. The distance was not too great for Nattie to distinguish the
man as the English merchant.
Drawing himself up the lad shook his fist at the apparition. The
action brought his head above the hedge. Something moving on the
other side caught his eye, and he dodged back just as a man arose
to his feet within easy touch.
Breathless with amazement, Nattie crouched down, and parting the
roots of the hedge, peered through. The fellow was cautiously
moving toward the house. Something in his walk seemed familiar.
Presently he reached a spot where the moon's bright rays fell upon
him.
A stifled cry of profound astonishment, not unmingled with terror,
came from the lad's lips, and he shrank back as if with the intention
of fleeing. He thought better of it, however, and watched with eager
eyes. A dozen times the man in the grounds halted and crouched to
the earth, but finally he reached the front entrance of the mansion.
A door was opened, and a hand was thrust forth with beckoning
fingers. The fellow hastily stepped inside and vanished from view,
leaving Nattie a-quiver with excitement. The dislocated shoulder, the
pain, the fever, all were forgotten in the importance of the discovery.
"That settles it," he muttered. "I am on the right track as sure as the
moon is shining. Now I must enter that house by hook or crook. But
who would believe that miracles could happen in this century? If that
fellow wasn't——"
He abruptly ceased speaking. The door in the front entrance
suddenly opened, and a huge dog was thrust down the stone steps.

Nattie knew the animal well. It was a ferocious brute Ralph had
imported from England that year.
As a watchdog it bore a well-merited reputation among the natives
of thieving propensities. It was dreaded because it thought more of
a direct application of sharp teeth than any amount of barking. Its
unexpected appearance on the scene altered matters considerably.
"Dog or no dog, I intend to find my way into that house before many
minutes," decided the lad. "It is an opportunity I cannot permit to
pass."
He drew out his revolver, but shook his head and restored it again to
his pocket. A shot would alarm the neighborhood and bring a squad
of police upon the scene. The brute must be silenced in some other
manner.
Naturally apt and resourceful, it was not long before Nattie thought
of a plan. Cautiously edging away from the hedge until he had
reached a safe distance, he set out at a run toward home.
Fortunately, the street was free from police or pedestrians, and he
finally gained the Manning residence without being observed.
Slipping into the garden he whistled softly. A big-jointed, lanky pup
slouched up to him and fawned about his feet. Picking up the dog,
he started back with it under his right arm. The return to the English
merchant's house was made without mishap.
Reaching the hedge, Nattie lightly tossed the pup over into the yard.
It struck the ground with a yelp, and a second later a dark shadow
streaked across the lawn from the mansion. As the lad had
anticipated, the dog he had brought did not wait to be attacked, but
started along the inner side of the hedge with fear-given speed. In
less than a moment pursuer and pursued disappeared behind an
outlying stable.
Chuckling at the success of his scheme, Nattie softly climbed the
fence and leaped into the yard. The lawn was bright with the rays of

the moon, but he walked across it without hesitation, finally reaching
the house near the left-hand corner.
As he expected, he found a side door unguarded save by a wire
screen. A swift slash with a strong pocket-knife gave an aperture
through which the lad forced his hand. To unfasten the latch was the
work of a second, and a brief space later he stood in a narrow hall
leading to the main corridor.

CHAPTER XXI.
A PRISONER.
On reaching the main stairway he heard voices overhead. The sound
seemed to come from a room opening into the hall above. Quickly
removing his shoes, the lad tied the strings together, and throwing
them about his neck, he ascended to the upper floor.
Fortunately, Nattie had visited the Black mansion in his earlier days
when he and Ralph were on terms of comparative intimacy. He knew
the general plan of the house, and the knowledge stood him in good
stead now.
The room from which the sound of voices came was a study used by
the English merchant himself. Next to it was a spare apartment filled
with odd pieces of furniture and what-not. In former days it was a
guest chamber, and the lad had occupied it one night while on a visit
to the merchant's son.
He remembered that a door, surmounted by a glass transom, led
from the study to the spare room, and that it would be an easy
matter to see into the former by that means.
He tried the knob, and found that it turned at his touch. A slight
rattle underneath proclaimed that a bunch of keys was swinging
from the lock. Closing the door behind him, he tiptoed across the
apartment, carefully avoiding the various articles of furniture.
To his great disappointment, he found that heavy folds of cloth had
been stretched across the transom, completely obstructing the view.
To make it worse, the voices were so faint that it was impossible for
him to distinguish more than an occasional word.

"Confound it! I have my labor for my pains!" he muttered. "It's a
risky thing, but I'll have to try the other door."
He had barely reached the hall when the talking in the next room
became louder, then he heard a rattling of the knob. The occupants
were on the point of leaving the study. To dart into the spare room
was Nattie's first action. Dropping behind a large dressing-case, he
listened intently.
"Well, I am thoroughly satisfied with your part of the affair so far,"
came to his eager ears in the English merchant's well-known voice.
"It was well planned in every respect. You had a narrow escape
though."
A deep chuckle came from the speaker's companion.
"No suspicion attaches to me," continued Mr. Black. "I met the boys
last night, but I don't think they saw me."
"Oh, didn't we?" murmured Nattie.
"You can go now. Give this letter of instructions to my son, and tell
him to make all haste to the place mentioned. Return here with his
answer as quickly as you can. In this purse you will find ample funds
to meet all legitimate expenses. Legitimate expenses, you
understand? If you fall by the wayside in the manner I mentioned
before you will not get a sen of the amount I promised you. Now—
confound those rascally servants of mine! they have left this room
unlocked! I must discharge the whole lot of them and get others."
Click! went the key in the door behind which Nattie crouched. He
was a prisoner!
The sound of footsteps came faintly to him; he heard the front
entrance open; then it closed again, and all was silent in the house.
After waiting a reasonable time he tried the knob, but it resisted his
efforts. Placing his right shoulder against the wood he attempted to
force the panel, but without avail.

"Whew! this is being caught in a trap certainly! A pretty fix I am in
now. And it is just the time to track that scoundrel. Mr. Black must
have been talking about poor Grant."
Rendered almost frantic by his position, Nattie threw himself against
the door with all his power. The only result was a deadly pain in the
injured shoulder. Almost ready to cry with chagrin and anguish, he
sat down upon a chair and gave himself up to bitter reflections.
Minutes passed, a clock in the study struck three; but still he sat
there a prey to conflicting emotions. He now saw that he had acted
foolishly. What had he learned? They had suspected the Blacks
before, and confirmation was not needed.
The discovery of the visitor's identity was something, but its
importance was more than counterbalanced by the disaster which
had befallen Nattie. The recent conversation in the hall indicated
that the merchant's companion would leave at once for a rendezvous
to meet Ralph, and possibly Grant.
"And here I am, fastened in like a disobedient child," groaned the
lad. "I must escape before daylight. If I am caught in here Mr. Black
can have me arrested on a charge of attempted burglary. It would
be just nuts to him."
The fear of delay, engendered by this new apprehension, spurred
him to renewed activity. He again examined the door, but speedily
gave up the attempt. Either a locksmith's tools or a heavy battering-
ram would be necessary to force it.
Creeping to the one window opening from the apartment, Nattie
found that he could raise it without much trouble. The generous rays
of the moon afforded ample light. By its aid he saw that a dense
mass of creeping vines almost covered that side of the mansion.
"By George! a chance at last!"
Cautiously crawling through the opening he clutched a thick stem
and tried to swing downward with his right hand. As he made the

effort a pain shot through his injured shoulder so intense that he
almost fainted. He repressed a cry with difficulty.
Weak and trembling, he managed to regain the window sill. Once in
the room he sank down upon the floor and battled with the greatest
anguish it had ever been his lot to feel.
To add to his suffering, came the conviction that he would be unable
to escape. He remembered the telltale slit he had made in the
screen door. When daylight arrived it would be discovered by the
servants, and a search instituted throughout the house.
"Well, it can't be helped," mused the lad. "If I am caught, I'm
caught, and that's all there is about it."
It is a difficult thing to philosophize when suffering with an intense
physical pain and in the throes of a growing fever. It was not long
before Nattie fell into a stupor.
He finally became conscious of an increasing light in the room, and
roused himself enough to glance from the window. Far in the
distance loomed the mighty volcano of Fuji San, appearing under the
marvelous touch of the morning sun like an inverted cone of many
jewels.
A hum of voices sounded in the lower part of the house, but no one
came to disturb him. Rendered drowsy by fever, he fell into a deep
slumber, and when he awoke it was to hear the study clock strike
nine. He had slept fully five hours.
Considerably refreshed, Nattie started up to again search for a way
to effect his escape. The pain had left his shoulder, but he felt an
overpowering thirst. His mind was clear, however, and that was half
the battle.
"If I had more strength in my left arm I would try those vines once
more," he said to himself. "Things can't last this way forever. I must
—what's that?"

Footsteps sounded in the hall outside. They drew nearer, and at last
stopped in front of the spare-room door. A hand was laid upon the
knob, and keys rattled.
"We have searched every room but this," came in the smooth tones
of the English merchant. "Go inside, my man, and see if a burglar is
hiding among the furniture. Here, take this revolver; and don't fear
to use it if necessary."
Like a hunted animal at bay, the lad glared about him. Discovery
seemed certain. Over in one corner he espied a chest of drawers. It
afforded poor concealment, but it was the best at hand. To drag it
away from the wall was the work of a second. When the door was
finally opened, Nattie was crouched behind the piece of furniture.
He heard the soft steps of a pair of sandals; he heard chairs and
various articles moved about, then the searcher approached his
corner. Desperate and ready to fight for his liberty, he glanced up—
and uttered a half-stifled cry of amazement and joy!

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Complex Manifolds 1, reprint with errata Edition James Morrow

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