Cusps of the Kähler moduli space and stability conditions on K3 surfaces

FourierMukai partners and stability conditions on K3
surfaces
Heinrich Hartmann
University of Bonn
24.2.2011
Heinrich Hartmann (University of Bonn) FourierMukai partners and stability conditions on K3 surfaces24.2.2011 1 / 19

Outline
1 Moduli spaces of sheaves on K3 surfaces
2 Stability conditions and the Kähler moduli space
3 Geometric interpretations of Ma's result
4 Future research plans

Moduli spaces on K3 surfaces are well behaved
Let X be a K3 surface and let N(X) = K(Coh(X))/rad(χ) be the
numerical K-group, endowed with the pairing (_._) = −χ(_, _).
Theorem (Mukai)
Let v ∈ N(X) be a vector with v.v = 0 and v.N(X) = Z. Then there
exists an ample class h ∈ NS(X) such that:
1 The moduli space M = Mh(v) is again a K3 surface.
In particular M is ne, smooth, compact and two-dimensional.
2 The FourierMukai functor is an equivalence:
ΦU : Db(M)
∼
−→ Db(X).

Ma interpretation of FourierMukai partners
Let X be a K3 surface. Recently, Shouhei Ma gave a surprising
interpretation of the set of FourierMukai partners of X:
Theorem (Ma)
There is a canonical bijection between
K3 surfaces Y
with Db(Y ) ∼= Db(X)
←→
standard cusps of the
Kähler moduli space KM(X)
.
There is a version for non-standard cusps and K3 surfaces twisted by
a Brauer class.
The proof uses deep theorems due to Mukai and Orlov to reduce the
statement to lattice theory.
Is there a geometric reason for this correspondence?

Construction of the Kähler moduli space
Consider the period domain
D(X) = { [z] ∈ P(N(X)C) | z.z = 0, z.¯z 0 } .
We and dene the Kähler moduli space to be
KM(X) = D(X)+
/Γ,
where Γ is the image of Aut(Db(X)) in O(N(X)).
The BailyBorel compactication KM(X) ⊂ KM(X) is a normal
projective variety. The complement KM(X) KM(X) consists of
components of dimension 0 and 1 which are in bijection to Bi/Γ, where
Bi = { I ⊂ N(X) | primitive, isotropic, rk(I) = i + 1 }
for i = 0, 1 respectively.
Boundary components of dimension 0 are called cusps.
We call a cusp [I] ∈ KM(X) standard if I.N(X) = Z.

Picture of the Kähler moduli space
Figure: Kähler moduli space with cusps, associated K3 surfaces and two dierent
degenerating paths.

Bridgeland stability conditions
Let D be a C-linear, triangulated category.
Denition (Bridgeland)
A stability condition σ on D consists of:
a heart of a bounded t-structure A ⊂ D and
a vector z ∈ N(D)C, called central charge.
satisfying the following properties:
1 For all E ∈ A, E = 0 the complex number (z.[E]) = r exp(iπφ)
satises r 0 and φ ∈ (0, 1].
2 Existence of Hader-Narasimhan ltrations.
3 Local niteness.
An object E ∈ A is called σ-stable if for all sub-objects F ⊂ E in A
φ(F) φ(E).

Stability conditions on K3 surfaces
Let X be a K3 surface and D = Db(X).
Theorem (Bridgeland)
The set of stability conditions Stab(D) on D has the structure of a
complex manifold. The map σ = (A, z) → z induces a Galois-cover
π : Stab†
(X) −→ P+
0 (X) ⊂
open
N(X)C,
where Stab†(X) is the connected component of Stab(D), containing the
stability conditions σX(ω, β). Moreover
Deck(π) ∼= Aut†
0(D)
is the group of auto-equivalences respecting the component Stab†(X) and
acting trivially on H∗(X, Z).

Stability conditions and the Kähler moduli space.
We get the following diagram
Stab†(X)
Gl+
2 (R)

π // P+
0 (X)
Gl+
2 (R)

Stab†(X)/Gl
+
2 (R) //
Aut†(D)

D+
0 (X)
Γ

Aut†(D) Stab†(X)/Gl
+
2 (R)
π // KM0(X),
where KM0(X) ⊂ KM(X) is a special open subset, and Aut†(D) ⊂ Aut(D)
is the subgroup of auto-equivalences which respect the distinguished
component.
Fact: π is an isomorphism.
This fact has been stated by Bridgeland and Ma before. However, it seems
to depend on the following results.

Equivalences respecting the distinguished component
Theorem
The following equivalences respect the distinguished component.
For a ne, compact, two-dimensional moduli space of Gieseker-stable
sheaves Mh(v), the FourierMukai equivalence induced by the
universal family.
The spherical twists along Gieseker-stable spherical vector bundles.
The spherical twists along OC(k) for a (−2)-curve C ⊂ X and k ∈ Z.
This allows us to show the following strengthening of a result of
[HuybrechtsMacriStellari].
Corollary
The map Aut†(Db(X)) −→ Γ ⊂ O(N(X)) is surjective.

Main question
There is a canonical map
¯π : Stab†
(X) −→ KM(X).
What is the relation between stability conditions σ with ¯π(σ) near
to a cusp and the associated K3 surface Y ?
1 How is the heart of σ related to the heart Coh(Y )?
2 Can we construct Y as a moduli space of σ-stable objects?

Cusps and hearts
We can always nd a degeneration σ(t) of stability conditions, such that
the hearts converge to Coh(Y ):
Theorem
Let [I] ∈ KM(X) be a standard cusp and Y the K3 surface associated to
[I] via Ma's theorem. Then there exists a path σ(t) ∈ Stab†(X), t 0 and
an equivalence Φ : Db(Y )
∼
−→ D such that
1 lim
t→∞
π(σ(t)) = [I] ∈ KM(X) and
2 lim
t→∞
A(σ(t)) = Φ(Coh(Y )) as subcategories of D.
There are many other hearts that can occur as limits!
How can we classify all of them?

Linear degenerations
We dene a class of pahts γ(t) ∈ KM(X) called linear degeneration to
a cusp [I].
The prototypical example of a linear degeneration is ¯π(σY (β, tω)),
where σY (β, ω) is an explicit stability condition associated to
β, ω ∈ NS(X)R with ω ample, dened by Bridgeland.
Proposition
Let [v] ∈ KM(X) be a standard cusp and γ(t) ∈ KM(X) be a linear
degeneration to [v], then γ(t) is a geodesic converging to [v].
Conjecture
Every geodesic converging to [v] is a linear degeneration.
True in the Picard-rank one case
True if one works with BorelSerre compactication, c.f. [Borel-Ji].

Classication of linear degenerations
Theorem
Let [I] be a standard cusp of KM(X). Let σ(t) ∈ Stab†(X) be a path in
the stability manifold such that ¯π(σ(t)) ∈ KM(X) is a linear degeneration
to [I]. Let Y be the K3 surface associated to [v] by Ma. Then there exist
1 a derived equivalence Φ : Db(Y )
∼
−→ D,
2 classes β ∈ NS(Y )R, ω ∈ Amp(Y ) and
3 a path g(t) ∈ Gl
+
2 (R)
such that
σ(t) = Φ∗(σ∗
Y (β, t ω) · g(t))
for all t 0.
Moreover, the hearts of σ(t) · g(t)−1 are independent of t for t 0 and
can be explicitly described as a tilt of Coh(Y ).

Moduli of complexes on K3 surfaces
Let D be the derived category of a K3 surface X. For a stability condition
in the sense of Bridgeland σ ∈ Stab(D) and v ∈ N(D) we consider the
following moduli-space of semi-stable objects
Mσ
(v) = { E ∈ D | E σ-semi-stable, [E] = v } /even shifts.
This space has the structure of an Artin-stack of nite type due to results
by Lieblich and Toda. We prove the following result.
Theorem
If v ∈ N(X) is a vector with v.v = 0, v.N(X) = Z and σ ∈ Stab†(X) is
v-general stability condition, then:
1 The moduli space Mσ(v) is represented by a K3 surface X.
2 The universal family U ∈ Db(M × X) induces a derived equivalence
ΦU : Db(M)
∼
−→ Db(X).

Immediate open questions:
Are all geodesics to cusps linear degenerations?
Borel and Ji show, that our linear degenerations are the
EDM-geodesics in the BorelSerre compactication of D(X)/Γ.
Study bers of the morphism
D(X)/Γ
BS
−→ D(X)/Γ
BB
= KM(X).
Is the stability manifold connected?
Do all auto-equivalences of Db(X) preserve the component Stab†(X)?
Open cases are:
Unstable spherical vector bundles
Moduli spaces of simple bundles

Cusps on abelian surfaces
Can Ma's theorem be generalized to other (CalabiYau)-varieties?
Abelian surfaces A are the rst test case.
The stability manifold has been described by Bridgeland:
Stab†
(A) −→ P+
(A)
is the universal cover.
The auto-equivalences of abelian varieties known by Orlov and
Polishchuk
0 −→ Z ⊕ (A × Â) −→ Aut(Db(A)) −→ U(A × Â) −→ 0,
where U(A × Â) ⊂ Aut(A × Â) is a certain explicit subgroup.

Automorphic functions on the stability manifold
The Kähler moduli space
KM0(X) = Aut†
(X) Stab†
(X)/Gl
+
2 (R)
is a quasi-projective variety. Sections of an ample line bundle give rise
to automorphic functions on the stability manifold.
Use DonaldsonThomas/Joyce invariants DTα(v) to construct
interesting functions on the stability manifold. (c.f. Toda,
MellitOkada)
Study Fourier-expansion of these functions at various cusps.
Already interesting in Picard-rank one case, where
KM(X) ∼= H/Γ+
0 (n)
is a Fricke modular curve.

Periods and Fano manifolds
In an other work we computed the mirror map
φ : KM(X)
∼
−→ CM(Y )
between the Kähler moduli space of a generic quartic X ⊂ P3 and the
complex deformation space of the mirror K3 surface (Dwork pencil).
We have KM(X) ∼= H/Γ+
0 (2) and CM(Y ) ∼= P1 {0, 1, ∞}, therefore
φ gives rise to a modular function ˜φ : H −→ C.
The function ˜φ is explicitly given as a quotient of solutions to the
PicardFuchs equation ˜φ = W1/W2.
By Mirror symmetry for the Fano manifold P3 the the PicardFuchs
equation for Y equals the Quantum dierential equation for P3.
The solutions W1, W2 can be constructed directly in terms of
GromovWitten invariants. (KatzarkovKontsevichPantev, Iritani)

Cusps of the Kähler moduli space and stability conditions on K3 surfaces

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