key: cord-0436802-ia1ad7ua
authors: Izadi, Elham; Canning, Samir; Dutta, Yajnaseni; Stapleton, David
title: Hyperk"ahler manifolds
date: 2021-12-04
journal: nan
DOI: nan
sha: 9ff3b782c2a8cf63eeff6994839ad2bcca92ab7a
doc_id: 436802
cord_uid: ia1ad7ua

We give an elementary introduction to hyperk"ahler manifolds, survey some of their interesting properties and some open problems.

The cohomology of a compact Kähler manifold has remarkable properties, abstractified in the modern notion of a (polarized) Hodge structure. While the datum of a Hodge structure of weight 1 is equivalent to the datum of a compact complex torus, this is no longer the case in higher weights. In weight 2 there are remarkable examples of compact Kähler manifolds which are, mostly, determined by the polarized Hodge structure on their second cohomology. These are the hyperkähler manifolds: higher dimensional analogues of K3 surfaces. In these lecture notes, we give an elementary introduction to hyperkähler manifolds and survey some of their interesting properties. We start by reviewing the notions of tensors, connections, the curvature tensor, Ricci curvature and some of their properties. We define parallel transport, holonomy and the Levi-Civita connection. We also describe the constraints posed by the holonomy on the curvature tensor. We define (locally) symmetric spaces and state the main structure theorem for them. We then state De Rham's decomposition theorem for simply connected complete Riemannian manifolds and Berger's classification of the holonomy groups of nonsymmetric, complete, connected, irreducible Riemannian manifolds. Berger's classification shows that hyperkähler manifolds are the nonsymmetric complete It is the content of the Torelli theorem that hyperkähler manifolds are essentially determined by their second cohomology. This is consistent with the fact that all constructions to date of hyperkähler manifolds involve surfaces.

We briefly describe the moduli spaces of compact hyperkähler manifolds, their period domains and some of their properties. By a result of Tian-Todorov and Bogomolov, the deformations of hyperkähler manifolds are unobstructed. This essentially means that the moduli spaces of compact hyperkähler manifolds are smooth analytic spaces. It is known however, that they are not Hausdorff. For a fixed compact hyperkähler X, we describe the local and the global period domains with their respective maps from the local and global deformation spaces of X. We explain the local Torelli theorem and Verbitsky's weaker version of global Torelli which holds in the hyperkähler case.

We conclude with a brief discussion of twistor conics and twistor families, the proof of the global Torelli theorem by Verbitsky and the relation between twistor families and hyperholomorphic bundles.

Some good general references for the material that we present here are: [Bea83] , [Bea07] , [Bea11] , [doC92] , [GHJ03] , [VK99] .

This is an expanded version of the notes for a series of ten 45 minute lectures that I gave at the Trieste algebraic geometry summer school in July 2021. Exercise sessions for the lectures were run by Samir Canning, Yajnaseni Dutta and David Stapleton whom I wish to thank for their help.

I also wish to thank the organizers: Valentina Beorchia, Ada Boralevi and Barbara Fantechi, for the invitation to lecture at the summer school and for the excellent organization of an enjoyable summer school, especially with the challenge of COVID-19.

1. C ∞ manifolds 1.1. Tangent and cotangent bundles. For a C ∞ manifold M , we denote by T M the tangent bundle of M and T * M the cotangent bundle. For any non-negative integers (k, l), the sections of the bundle T ⊗k M ⊗ (T * M ) ⊗l are called (k, l)tensors. Section of T M are vector fields and sections of Λ p T * M differential p-forms. Alternatively, vector fields can be defined as first order differential operators on C ∞ functions.

In a local coordinate chart with local coordinates (x 1 , . . . , x n ), the (local) vector fields ∂/∂x 1 , . . . , ∂/∂x n form a basis of vector fields and the (local) 1-forms dx 1 , . . . , dx n form a basis of differential 1-forms.

A local (k, l)-tensor can be written as

1.2. The Lie bracket. Given a vector field v = v i ∂ ∂x i and a C ∞ function f on M ,

Given two vector fields v = v i ∂ ∂x i , w = w i ∂ ∂x i , the Lie bracket of v and w is given by

Alternatively, the Lie bracket can be defined via its action on C ∞ functions on M :

1.3. Connections. Tangent vectors allow us to take derivatives of C ∞ functions. Connections allow us to take derivatives of sections of arbitrary vector bundles.

For a C ∞ vector bundle E on M , a connection is a linear map

satisfying the Leibnitz rule

for all C ∞ sections e of E and C ∞ functions f on M . For any vector field v on M , the connection

We call ∇ v the covariant derivative in the direction of v.

We may thus also think of ∇ as a linear map

When E = T M , the torsion of a connection ∇ :

We say ∇ is torsion-free or symmetric when T = 0. Precisely, the curvature of a connection ∇ is, a linear map

or, equivalently,

or a global section

. It can be defined via its action on sections e of E and vector fields v, w as

We say that the connection ∇ (or sometimes the bundle E) is flat if R = 0.

In a coordinate chart with coordinates (x 1 , . . . , x n ), the partial derivatives commute, i.e.,

∇ ∂ ∂x i (e) and the connection is flat if and only if its partial (covariant) derivatives commute.

1.5. Parallel transport. Suppose given a C ∞ vector bundle E on M with a connection

and a smooth curve γ : [0, 1] → M . Parallel transport along γ produces sections of the pullback γ * E that are 'constant' or 'horizontal' along γ. As we see below, such sections exist and are determined by their values at one point of γ.

The pull-back γ * E is a C ∞ vector bundle on [0, 1] with fiber E γ(t) at t ∈ [0, 1]. The connection ∇ defines the connection γ * ∇ on γ * E as the composition

where the second map is induced by the projection T * M → → T * [0,1] . In local coordinates (x 1 , . . . , x n ) on M , with γ(t) = (x 1 (t), . . . , x n (t)),

and, for all (local) sections e of E,

Definition and Proposition 1.1. Put x := γ(0), y := γ(1). Then, for all e ∈ E x = (γ * E) 0 , there exists a unique smooth section s of γ * E such that s(0) = e and γ * ∇(s) = 0, i.e., ∇γ (t) (s) = 0.

The parallel transport of e along γ to y is P γ (e) := s(1) ∈ E y = (γ * E) 1 . The map

1.6. Holonomy. As we saw above, parallel transport defines linear isomorphisms between fibers of E at points of M . In particular, for a given point x of M , it defines linear automorphisms of the fiber E x . The holonomy of ∇ is the group generated by these automorphisms. It acts on all tensors of E and its invariants are the covariantly constant tensors:

Definition and Proposition 1.2. If γ is a loop (i.e. x = y), then P γ ∈ GL(E x ). The holonomy

group Hol x (∇) at x is

It has the following properties.

(1) Hol x (∇) is a Lie subgroup of GL(E x ):

(2) If γ is a path from x to y, then Hol y (∇) = P γ Hol x (∇)P −1 γ .

Hence, up to conjugation, Hol x (∇) only depends on the connected component of M containing x.

(3) if M is simply connected, then Hol x (∇) is connected. Any loop can be shrunk to a point:

Then {P s := P γs | s ∈ [0, 1]} is a path in Hol x (∇) from P 0 = P γ 0 to P 1 = P γ 1 = Id.

(4) Let hol x (∇) ⊂ gl(E x ) = End(E x ) be the Lie algebra of Hol x (∇). Recall that the curvature

As we shall see below, Riemannian holonomy plays a central role in the structure theory of Riemannian manifolds.

The connection ∇ induces connections on all tensor powers E ⊗k ⊗ (E * ) ⊗l , and all exterior and symmetric powers of E and E * and their tensor products. We shall denote these induced connections by ∇ as well.

Theorem 1.4. For a tensor S, ∇(S) = 0 if and only if S is fixed by Hol x (∇), if and only if P γ (S(x)) = S(y) for all x, y ∈ M and all paths γ from x to y.

A C ∞ manifold is called Riemannian if it has a Riemannian metric, i.e., a (2,0)-tensor g ∈

and defines a positive definite quadratic form on the tangent space T M,x for all x ∈ M . It is a fundamental result in differential geometry that every smooth manifold can be endowed with a Riemannian metric.

Riemannian manifolds have canonical connections on their tangent bundles: the Levi-Civita connection. The holonomy of the Levi-Civita connection is called Riemannian holonomy and the classification of Riemannian manifolds is based on the classification of Riemannian holonomy groups.

2.1. Levi-Civita connection. Suppose (M, g) is a Riemannian manifold. The fundamental theorem of Riemannian geometry is the following.

Theorem 2.1. There exists a unique torsion free (or symmetric) connection ∇ on T M such that ∇g = 0. This unique connection is called the Levi-Civita or Riemannian connection of (M, g).

One can verify that the condition ∇g = 0 is equivalent to the following compatibility property:

For all vector fields u, v, w on M ,

The Levi-Civita connection ∇ can be explicitly defined via

The curvature R(∇) is a (1, 3) tensor:

More symmetries of R(∇) can be exhibited by defining the (0, 4) tensor R(∇) as the compostion

While a priori R(∇) ∈ C ∞ ((T * M ) ⊗2 ⊗ Λ 2 T * M ), one can show that in fact

The Bianchi identities can be written in the form

In a basis of local coordinates x 1 , . . . , x n , we can write R(∇) as

where α β := α ⊗ β + β ⊗ α is the symmetric tensor. The Bianchi identities then can be written as

The Ricci curvature is a (0, 2) tensor, obtained by contracting R(∇):

At each point x ∈ M , the curvature tensor R defines a multilinear map

The Ricci curvature is the (0, 2) tensor defined as

where tr is the trace of a linear map. It follows from the symmetries of the curvature tensor that the Ricci curvature is symmetric. In local coordinates, if we write the curvature tensor as 

A first symmetry property of Riemannian holonomy is seen using the isomorphism g :

Proposition 2.3. We have

We saw that the curvature tensor R ∈ (hol

2.4. Reducibility. The first step in the classification of Riemannian manifolds is to decompose them into their 'irreducible' factors. As we see below, these correspond to the irreducible summands in the representation of the Riemannian holonomy group on the tangent space of M .

Definition 2.5. A Riemannian manifold is called (locally) reducible if every point has a neighborhood isometric to a product. It is called irreducible if it is not locally reducible. We have Proposition 2.6. Suppose a neighborhood of x ∈ M is isometric to the product (M 1 , g 1 )×(M 2 , g 2 ).

Then Hol x (g 1 × g 2 ) = Hol x (g 1 ) × Hol x (g 2 ).

Theorem 2.7. If (M, g) is irreducible at x, then R n = T x M is an irreducible representation of Hol x (g).

2.5. Symmetric and locally symmetric spaces. A large and relatively well understood class of irreducible Riemannian manifolds is that of locally symmetric spaces. 2.6. Geodesics and completeness. To better understand locally symmetric spaces, we use 'geodesics'.

Geodesics allow us to define a notion of 'completeness' (often called geodesic completeness) for Riemannian manifolds. Among other things, these notions allow us to describe all symmetric spaces in terms of Lie groups.

Definition 2.11. A geodesic is a parametrized smooth curve γ : (a, b) → M such that, for all t ∈ (a, b), ∇γ (t)γ (t) = 0.

Intuitively, a geodesic is the trajectory of a particle moving with constant velocity on the manifold:

the equation ∇γ (t)γ (t) = 0 means that the acceleration of the particle is 0 with respect to the Levi-Civita connection.

The Riemannian metric defines a norm in the tangent space at each point of M . By integrating the length of the velocity vector of a parametrized (piecewise) smooth curve, we define the length of such a curve. One can show that geodesics are locally the 'shortest' curves on M for the Riemannian length. It can happen however that there are many geodesics of different lengths between two given points on a manifold. The simplest example of this is the cylinder with Riemannian metric induced from R 3 . The Riemannian distance is defined as the infimum of the lengths of the (piecewise)

smooth curves between two points on M . We have the following useful existence and uniqueness theorem for geodesics. 

Riemannian manifold which is also a complete metric space with respect to the Riemannian distance is geodesically complete.

We can now give the description of symmetric spaces in terms of Lie groups.

Proposition 2.14. Suppose (M, g) is a connected, simply connected symmetric space. Then (M, g)

is complete. Put

Then G is a connected Lie group. Choose p ∈ M and let H be the stabilizer subgroup of p in G.

Then H is a closed connected Lie subgroup of G and the map

is a diffeomorphism. Berger's theorem classifies the possibilities for the restricted holonomy group Hol(g) 0 and describes the corresponding manifolds.

Theorem 2.17. Suppose (M, g) is Riemannian, complete, connected, nonsymmetric, irreducible.

Then the restricted holonomy group Hol(g) 0 is one of the following:

(1) Hol(g) 0 ∼ = SO(n) (automorphisms of R n preserving the metric, generic metric),

(2) n = 2m ≥ 4, Hol(g) 0 = U (m) ⊂ SO(n) (automorphisms of C m perserving a hermitian form, Kähler),

Kähler), 

For a complex manifold M , multiplication by i defines an endomorphism I : T M → T M satisfying

The (1, 1) form associated to g and I is ω(v, w) := g(Iv, w), for all vector fields v, w.

Equivalently, ω is the composition

The fact that ω is a (1, 1) form means ω(Iv, Iw) = ω(v, w). One also checks that ω is anti-symmetric.

It is easy to check that any two of {I, g, ω} determine the third.

Definition and Proposition 3.1. The metric g is Kähler with respect to I if one of the following equivalent conditions hold:

(1) dω = 0,

(2) ∇ω = 0,

In such a case, ω is called the Kähler form of g.

So g is Kähler if and only if ω and I are constant. Equivalently Hol(g) preserves ω and I.

The subgroup of SO(n) preserving I is U (m) (n = 2m). Therefore, M is Kähler if and only if

3.1. Ricci form. Given a Kähler manifold (M, g, I), its Ricci form ρ is the differential form associated to the Ricci curvature via I:

Equivalently, ρ is the composition

As in the case of ω and g: ρ ∈ C ∞ (Λ 2 T * M ). We have the following 

is an isomorphism.

Hence K M has a nowhere vanishing holomorphic section, which implies that K M is trivial, i.e.,

M is Calabi-Yau. Furthermore, on the tangent space T p M at a point p ∈ M , a nonvanishing differential m-form is a multiple of the determinant. Hence Hol(g) preserves the determinant. Since we already know that Hol(g) ⊂ U (m), this implies that Hol(g) ⊂ SU (m).

Conversely, if Hol(g) ⊂ SU (m), then M admits a nowhere vanishing differential m-form, K M is trivial and ρ = 0.

3.3. The hyperkähler case. Recall that the quaternions have bases of the form

A triple (i, j, k) as above is called a quaternionic triple. The Lie group Sp(r) is the group of R-linear endomorphisms of H r preserving a quaternionic Hermitian form q. Recall that q is quaternionic

We can embed Sp(r) in SU (2r) each time we choose i ∈ H with i 2 = −1 as follows.

Complete i to a quaternionic triple (i, j, k) and write

where h is Hermitian with respect to i and ω is alternating C-bilinear with respect to the complex structure on H r given by i. Then Sp(r) can be identified with the group of R-linear automorphisms of H preserving h and ω. Hence, thinking of U (2r) as the group of transformations of H r = C ⊕ Ci preserving h, we can identify Sp(r) as the subgroup of U (2r) of transformations preserving ω. In particular, they preserve ∧ r ω, which means they belong to SU (2r).

Given a Riemannian manifold M with Hol p (g) ⊂ Sp(r), we can identify T p M with H r . The form ω obtained as above by decomposing the form q is invariant under the holonomy group of M , hence globalizes to an alternating flat, i.e., holomorphic, 2-form on M which is non-degenerate everywhere. Furthermore, the quaternionic triple (i, j, k) gives three complex structures I, J, K on M satisfying the quaternionic relations and with respect to which g is Kähler (I, J, K are invariant under Hol p (g), hence flat). We then obtain a sphere of complex structures λ = aI + bJ + cK with a, b, c ∈ R, a 2 + b 2 + c 2 = 1 such that ∇λ = 0. The metric g is therefore Kähler with respect to all these complex structures. If M is a complex torus, then Hol(g) = 0. Any complex structure is then Kähler.

Definition 3.4. We say that M is irreducible hyperkähler if Hol(g) = Sp(r), i.e., M has exactly an S 2 of Kähler complex strcutures.

3.4. The Calabi conjecture and its consequence. [ρ] = 2πc 1 (K M ). There exists a unique Kähler metric g on M whose Ricci form is ρ and whose

For Ricci-flat manifolds this has the following useful consequence. (1) the universal cover of M is isomorphic to C k × i V i × j X j where C k has the standard Kähler metric, and, for all i, V i is compact simply connected with holonomy SU (m i ) and, for all j, X j is compact simply connected with holonomy Sp(r j ),

Lemma 3.8. Suppose (M, I, g) is a compact Kähler, simply connected, Ricci-flat manifold. The group of automorphisms of (M, I) is discrete. In particular, the group of automorphisms of (M, I, g)

is finite (because it is contained in SO(n) which is compact).

We now present the infinite series of examples of compact hyperkähler manifolds constructed by Beauville [Bea83] . For this, the point of view of holomorphic symplectic geometry is more convenient. We begin with the following. (2) for all 0 ≤ p ≤ r,

Definition and Proposition 4.2. A compact Kähler manifold X is called holomorphic symplectic if there exists an everywhere non-degenerate holomorphic 2-form on X. This is equivalent to: X is compact hyperkähler or X is Kähler and Hol g (X) ⊂ Sp(r).

A compact Kähler manifold X is called irreducible holomorphic symplectic if X is simply connected and H 2 (X, Ω 2 X ) is generated by an everywhere non-degenerate holomorphic 2-form. This is equivalent to: X is irreducible compact hyperkähler X is Kähler and Hol g (X) = Sp(r).

4.1. The case of surfaces. In dimension 2, Sp(1) = SU (2), so Calabi-Yau and hyperkähler are the same: these are K3 surfaces and complex tori.

One can prove that K3 surfaces are simply connected and their integral cohomology is torsion free.

It is a deep theorem of Siu that a K3 surface admits a unique Kähler metric.

Examples of algebraic K3 surfaces:

(1) Double covers of P 2 branched along smooth sextics.

(2) Smooth quartics in P 3 .

(3) (2, 3) complete intersections in P 4 .

(4) (2, 2, 2) complete intersections in P 5 . (2) The complex manifold S

[r] * is the blow up of S (r) * along D * .

(3) If we denote Bl ∆ (S r * ) the blow up of S r * along the union of its diagonals, then the action of S r lifts to Bl ∆ (S r * ) and S [r] * = Bl ∆ (S r * )/S r .

Next we construct differential forms on S [r] , starting from differential forms on S.

Given a holomorphic differential form ω on S, the form ψ := pr * 1 ω + . . . + pr * r ω and its pull-back η * ψ to Bl ∆ (S r * ) are invariant under the action of S r . Hence there exists a holomorphic differential form ϕ on S

[r] * such that η * ψ = ρ * ϕ.

Proposition 4.5. If K S = Ω 2 S is trivial, then S [r] admits a holomorphic symplectic form.

Proof. Let ω be a generator of K S . Defining ψ and ϕ as above, we show that ϕ extends to S [r] as an everywhere non-degenerate form.

The form ϕ extends to all of S [r] because S [r] \ S

[r] * has codimension ≥ 2 in S [r] . The fact that ϕ is everywhere non-degenerate means that ∧ r ϕ does not vanish anywhere.

The form ∧ r ϕ is a section of K S [r] , so the locus where it vanishes is a canonical divisor on S [r] .

Denote E ij := η * ∆ ij . Then the divisors E ij are the exceptional divisors of the blow up η :

Bl ∆ (S r * ) → S r * and the ramification divisors of the morphism ρ :

and the divisor of zeros of ρ * ∧ r ϕ is

However,

Indeed, choose z = (x 1 , . . . , x r ) ∈ S r , then

The differential form ψ is a bilinear form on T z S r , the decomposition T z S r = T x 1 S ⊕ . . . ⊕ T xr S is orthogonal with respect to ψ and ψ is non-degenerate at any z. Hence Div(∧ r ψ) = 0 on S r . However, the differential of the blow up η : Bl ∆ (S r * ) → S r * has image of dimension 2r − 1 along the union of the diagonals, so η * ψ is degenerate of rank 2r − 2 along ∪ i<j E ij . It follows that

So ρ * Div(∧ r ϕ) = 0 and Div(∧ r ϕ) = 0.

To determine the type of S [r] , we compute its fundamental group. The map S r → S (r) is a Galois cover with Galois group S r . So we have the exact sequence of fundamental groups 1 −→ S r −→ π 1 (S r ) −→ π 1 (S (r) ) −→ 1.

We have

The map BL ∆ (S r * ) → S

[r] * is also a Galois cover with Galois group S r . So we have the commutative diagram of exact sequences

Therefore, we also have π 1 (S

It is a fact from algebraic topology and group theory that π 1 (S (r) ) is the largest commutative quotient of π 1 (S), hence it is isomorphic to H 1 (S, Z).

Lemma 4.6.

(1)

(3)

Proof.

(1) Standard.

(2) Replace S r by S r * , S (r) by S (r) * and S [r] by S

[r] * : the second cohomology does not change. We compute

(3) We compute, using part (1),

by skew-symmetry.

We immediately obtain. 

So in this case, the Hilbert scheme is not irreducible holomorphic symplectic. We determine its factors according to the decomposition theorem.

Consider the addition map s : A (r+1) → A and its composition

Definition 4.8. The (r + 1)-st generalized Kummer manifold of A is

One can see that K r is a manifold as follows.

The complex torus A acts on itself by translation, hence also on A [r+1] by pull-back:

If Z ⊂ A is an analytic subspace of length r + 1, then a ∈ A acts as Z → t * a Z on A [r+1] . The map S is equivariant for this action on A [r+1] and the action of A on itself via x → t * (r+1)a x. In other words we have the Cartesian diagram

It follows that S is a smooth map and all its fibers are isomorphic to K r which is therefore also smooth.

Proposition 4.9. The holomorphic symplectic structure of A [r+1] restricts to a holomorphic symplectic structure on K r .

Proof. Since K r is a fiber of a smooth morphism, its normal bundle is trivial: the normal space at

Recall the differential forms ψ = pr * 1 ω ⊕ . . . ⊕ pr * r+1 ω and ϕ with η * ψ = ρ * ϕ. The form ∧ r (ϕ| Kr ) is a section of K Kr ∼ = O Kr . We show that it remains everywhere non-degenerate. As before, this means that ∧ r (ϕ| Kr ) does not vanish anywhere. Since K Kr is trivial, either ∧ r (ϕ| Kr ) is zero everywhere or it does not vanish anywhere. We prove that it is nonzero at one point.

Let Z = x 1 + . . . + x r+1 ∈ K r be such that the x i are all distinct. Then

We can choose the isomorphism above in such a way that the differential dS : T Z A [r+1] → T 0 A of S is the sum map. The form ϕ acts as ω on each summand T 0 A of T Z A [r+1] and the summands are orthogonal to each for ϕ. It is then an exercise in linear algebra to check that ϕ| Ker dS is non-degenerate, i.e., ∧ r (ϕ| Kr ) is not 0.

Proposition 4.10. The manifold K r is simply connected. For r ≥ 2, we have

where E is the intersection of the exceptional divisor of A [r+1] with K r . Definition 5.1. A family of complex manifolds is a smooth proper morphism of complex spaces π : X → S.

A deformation of (X, I) is a family of complex manifolds with a point s 0 ∈ S and an isomorphism X 0 := π −1 (s 0 ) ∼ = X.

A deformation is called universal if, for any deformation X → S , there exists a unique morphism ϕ : S → S such that ϕ(s 0 ) = s 0 and X → S is the pull-back of X → S under ϕ. In other words, we have the Cartesian diagram

The universal deformation is unique up to unique isomorphism and we denote it X → Def(X).

Kuranishi's theorem is the following.

Theorem 5.2. Suppose (X, I) is a compaxt complex manifold with H 0 (X, T X ) = 0. Then a local universal deformation of (X, I) exists and it is universal for all of its fibers.

Under the conditions of the theorem, the local universal deformation X → Def(X) is sometimes called the Kuranishi family.

Note that the condition H 0 (X, T X ) = 0 means there are no global holomorphic vector fields on X or X has no infinitesimal automorphisms: given two complex manifolds X, Y and a holomorphic map f : X → Y , the tangent space to the space of holomorphic maps Hom(X, Y ) at f can be identified with H 0 (X, f * T Y ). This can be deduced from general results in deformation theory, applied to the deformations of the graph of f in X × Y . 5.3. Unobstructedness for K-trivial Kähler manifolds. For any compact complex manifold X, if H 0 (X, T X ) = 0, then X has a local or small universal deformation denoted X → Def(X).

By this we mean a germ of a deformation, i.e., whose base is suitably small. Such a deformation is universal for all its fibers, its base Def(X) is a "Kuranishi slice" ⊂ H 1 (X, T X ). For t ∈ Def(X) small, we have

The obstructions to deformations (to various orders) provide local analytic equations for Def(X) in a neighborhood of 0 ∈ H 1 (X, T X ). We say that the deformations of X are unobstructed if all the obstructions to deformations are 0.

If the deformations of X are unobstructed (i.e., dim T 0 Def(X) = dim Def(X)), then the base Def(X) is a small open neighborhood of the origin in H 1 (X, T X ). The following theorem is due to Bogomolov in the hyperkähler case and to Tian-Todorov in the general case.

Theorem 5.3. If the canonical bundle K X is trivial (we say X is K-trivial), then the deformations of X are unobstructed.

We have the following facts.

• If X is Kähler, then so is any small deformation of X.

• If X is Kähler and K-trivial, then small deformations X t of X are also Kähler and K-trivial

• If X is holomorphic symplectic, then small deformations of X are also holomorphic symplectic. If X is irreducible holomorphic symplectic, then all fibers of any deformation of X are irreducible holomorphic symplectic. We define the period domain using the second cohomology of hyperkähler manifolds, together with a non-degenerate quadratic form: the Beauville-Bogomolov form.

Suppose X is irreducible holomorphic symplectic (irreducible hyperkähler) of dimension 2n and choose σ ∈ H 0 (Ω 2 X ) such that X (σσ) n = 1.

For α ∈ H 2 (X, C), define

One can show this is equal to

Beauville showed that there exists d X ∈ N such that

Therefore, if r X is the positive real root of d X , then q X := r X q X is an n-th root of the n-th power cup-product on H 2 (X, C).

The quadratic form q X is integer valued on H 2 (X, Z), indivisible, non-degenerate, of signature

for t close to 0 in any deformation of X.

The form q is called the Beauville-Bogomolov form of the hyperkähler manifold. The inspiration for the Beauville-Bogomolov form came from the study of the Fano variety of lines of a cubic fourfold. There, it naturally appears as the intersection form on the fourth cohomology of the cubic threefold which is isomorphic to the second cohomology of its Fano variety of lines which is a hyperkähler manifold.

Note that for n = 1, q X = 2q X is the usual intersection form on H 2 (X, Z).

Define

We saw that for t ∈ Def(X) close to 0, q X (σ t ) = 0, q X (σ t + σ t ) > 0. Hence we can define the local period map P X : Def(X) −→ Q X This is holomorphic because σ t = H 2,0 (X t ) = H 0 (Ω 2 Xt ) varies holomorphically with t: H 0 (Ω 2 Xt ) is the fiber of the holomorphic line bundle π * Ω 2 X / Def(X) on Def(X). We have the local Torelli theorem:

Theorem 5.4. The local Torelli map P X is a local isomorphism, i.e., dP X is an isomorphism at 0. 5.6. The period domain. We now construct the global period domain for hyperkähler manifolds.

For this we first fix the discrete data of a lattice which will usually be abstractly isomorphic to the second integral cohomology of a hyperkähler manifold with its Beauville-Bogomolov form.

Definition 5.5. A lattice is the data of a free Z-module Γ of finite rank with an integral nondegenerate quadratic form q Γ .

Definition 5.6. Given a lattice (Γ, q Γ ), the period domain Q Γ is

The moduli space of marked holomorphic symplectic manifolds and local period maps. We will construct a moduli space of marked holomorphic symplectic manifolds and a global period map on it which is, roughly speaking, a glueing of local period maps.

Definition 5.7.

(1) A marking of an irreducible holomorphic symplectic manifold is a lattice isomorphism ϕ : (H 2 (X, Z), q X )

(2) The pair (X, ϕ) is called a marked manifold.

(3) Two marked manifolds (X, ϕ), (X , ϕ ) are isomorphic if there exists f : X → X such that ϕ = ϕ • f * . We write (X, ϕ) ∼ = (X , ϕ ).

(4) The moduli space of marked irreducible holomorphic symplectic manifolds is the set

We use the local period map to show that M Γ is a smooth (non-Hausdorff) complex analytic space:

Given an irreducible holomorphic manifold X, choose a marking ϕ : H 2 (X, Z) → Γ. The

Kuranishi family X → Def(X) is locally isomorphic to the period domain Q Γ : The marking ϕ : H 2 (X, Z) → Γ induces isomorphisms forming the commutative diagram Definition 5.8. The global period map is

Verbitsky's global Torelli theorem [Ver13] (also see [Huy12] and [Loo21] ) for compact hyperkähler manifolds is the following.

Theorem 5.9. The map P is generically injective on each connected component of M Γ .

Note that the datum of the line H 2,0 (X) ⊂ H 2 (X, C) determines the Hodge structure on

We say that the global Torelli theorem holds for a class of manifolds, if a manifold is determined In fact we have stronger Torelli theorems in the above cases: for complex tori, any Hodge isomorphism between the first cohomologies of two tori is induced by an isomorphism of the tori. For curves, any Hodge isometry between their first cohomologies is induced by an isomorphism between the curves up to a change of sign. For generic K3 surfaces, any Hodge isometry between the second cohomologies is induced by an isomorphism of the surfaces up to a sign.

For hyperkähler manifolds of dimension > 4, none of the above stronger versions of Torelli hold.

There are examples of (1) non-isomorphic (but bimeromorphic) compact hyperkähler manifolds with Hodge isometric second cohomologies [Deb84] ,

(2) non-birational projective hyperkähler manifolds of dimension 4 with Hodge isometric second cohomologies, [Nam02] . 

The spaces Teich(X) and M Γ (X) are non Hausdorff smooth analytic spaces and Q Γ is a (Hausdorff) simply connected complex manifold. Verbitsky constructed a new (Hausdorff) complex manifold M s Γ (X) which is obtained by identifying all non-separated points of M Γ (X). In other words

where, for two points p, q ∈ M Γ (X), p ≡ q when every neighborhood of p contains q and every neighborhood of q contains p. The period map then factors through M s Γ (X):

Theorem 5.11. The map P s Γ is surjective from any connected component of M s Γ (X) to Q Γ .

Combined with the facts that P s Γ is a local isomorphism and Q Γ is simply connected, this implies

Corollary 5.12. The map P s Γ induces an isomorphism from any connected component of M s Γ (X) to Q Γ .

Verbitsky's proof uses twistor conics which we will describe in the next section.

The following results of Huybrechts help us understand the difference between M Γ (X) and M s Γ (X).

Proposition 5.13. If two marked hyperkähler manifolds (X, ϕ) and (X , ϕ ) correspond to two non-separated points of M Γ (X), then X and Y are bimeromorphic and their period P Γ (X, ϕ) =

Proposition 5.14. Suppose given a bimeromorphism f : X → X between compact, hyperkähler manifolds. Then there exists families of compact hyperkähler manifolds

(2) there exists a bimeromorphism F : X → X commuting with the projections to D which is an isomorphism over D \ {0} and induces f on

Proposition 5.15. For any x ∈ Q Γ , the set of hyperkähler complex structures on a differentiable manifold X with period x ∈ Q Γ consists of a finite number of bimeromorphic equivalence classes.

6. Twistor spaces and twistor conics 6.1. Hyperkähler structures. Given X hyperkähler, let g be the hyperkähler metric of X. We saw that there exists complex structures I, J, K such that g is Kähler with respect to I, J, K and IJK = −1. In fact g is Kähler with respect to any linear combination λ = aI + bJ + cK such that a 2 + b 2 + c 2 = 1. The Kähler form associated to λ is ω λ (·, ·) := g(λ·, ·). So we have a family {(X, λ) | λ ∈ S 2 } of compact Kähler manifolds.

6.2. Twistor spaces. With the notation above, the twistor space X → P 1 of (X, g) is the product X × P 1 (as a real manifold) endowed with the almost complex structure

which is integrable by a result of Hitchin, Karlhede, Lindström, Roček.

6.3. Twistor conics. Fix a lattice (Γ, q Γ ), isometric to (H 2 (X, Z), q X ). Recall that the signature

where b 2 is the second Betti number of X. Since P 1 is simply connected, we can choose consistent markings on all the fibers of X → P 1 to obtain the period map

whose image is a twistor conic.

One can show that it is the intersection of a linearly embedded P = P 2 with Q Γ in P(Γ ⊗ C).

Furthermore P = P(W ⊗ C) where W is a three dimensional real subspace of Γ ⊗ R totally positive for the intersection form q Γ .

Conversely, one can show that each choice of a 3-dimensional real space W ⊂ Γ ⊗ R positive for q Γ gives a twistor conic:

Recall the following Definition 6.1. A Kähler class is the cohomology class of a (1, 1) form which is Kähler with respect to some metric. The Kähler cone is the cone generated by all Kähler classes.

A consequence of the Calabi-Yau theorem is the following. Intuitively, considering the twistor family

the C ∞ vector bundle B × P 1 on X has a structure of complex vector bundle holomorphic on each fiber (X, λ) of X → P 1 .

Stability conditions allow us to construct moduli spaces of bundles.

Definition 6.7. Fix a Kähler form ω on X. For a coherent sheaf F on X, put

where n is the complex dimension of X and vol(X) := X ω n . Define

where rank(F ) is the complex rank of F . We say F is stable with respect to ω if for all subsheaves

We say F is semi-stable with respect to ω if for all subsheaves F ⊂ F , we have slope(F ) ≤ slope(F ).

Verbitsky (see [VK99] ) proved that, given a vector bundle B on (X, I), if c 1 (B) and c 2 (B) are of type (1, 1) and (2, 2) with respect to all complex structures λ ∈ S 2 = P 1 on X, then B is hyperholomorphic. In particular, the class c 2 (B) is analytic on each (X, λ).

A useful characterization of stable bundles is given by the Hitchin-Kobayashi correspondence. To state it, we first need the following definition. When we write c i (X), we mean c i (T X ), where T X is the tangent bundle. Here are the first few terms of the Chern character and Todd class for reference:

and td(F ) = 1 + 1 2 c 1 (F ) 2 + 1 12 (c 1 (F ) 2 + c 2 (F )) + · · · Problem 7.4. Compute c 2 (X) for X a K3 surface. (Hint: set F = O X .)

Problem 7.5. Compute H 2 (X, Z). (Hint: take F = Ω X .)

You have now computed all of the Betti numbers. Next, we will compute the Hodge numbers.

Definition 7.6. Let X be a compact Kähler manifold. The Hodge numbers of X are h p,q = dim H q (X, Ω p X ).

Theorem 7.7 (The Hodge Decomposition). Let X be a compact Kähler manifold. There is a direct sum decomposition

Moreover h p,q = h q,p .

Problem 7.8. Compute all of the Hodge numbers of a compact complex K3 surface X. We begin with some basic problems.

Problem 7.10. Convince yourself that any holomorphic two-form σ on a complex manifold X induces a morphism of bundles

where T X is the tangent bundle and Ω 1 X is the cotangent bundle.

We call σ non-degenerate if the morphism above is an isomorphism.

Problem 7.11. Can you convince yourself that K3 surfaces are irreducible hyperkähler? (Hint: the tricky part is probably the simply connectedness. It may require some extra background knowledge.)

Problem 7.12. Show that h 2,0 = h 0,2 = 1, K X ∼ = O X , and that dim(X) is even for any irreducible compact hyperkähler manifold X.

Now that we know that K X is trivial for compact hyperkähler manifolds X, a natural question is: given a K X -trivial manifold, how can we show that it is hyperkähler, if it is? We will focus on a real-life example due to Debarre-Voisin [DV10] . The same type of argument works for another famous example of Beauville-Donagi [BD85] (the Fano variety of lines on a cubic fourfold.)

Let V 10 be a 10-dimensional complex vector space. Let ω ∈ ∧ 3 V ∨ 10 be a 3-form on V 10 . We define a subvariety of G(6, V 10 ):

Problem 7.13. Show that for a general choice of ω, X ω is a smooth fourfold. (Hint: show that X ω is given by the vanishing of a section of a certain globally generated vector bundle.)

Problem 7.14. Show that K Xω ∼ = O Xω . (Hint: use adjunction.)

Now that we know we have a K X -trivial variety, we want to show it's hyperkähler. Using something called the Koszul resolution, one can compute the Euler characteristic of the structure sheaf:

Definition 7.15. A strict Calabi-Yau manifold is a simply connected projective manifold X such that H 0 (X, Ω p X ) = 0 for 0 < p < dim(X).

Problem 7.16. Show that any simply connected smooth K X -trivial compact Kähler fourfold with χ(X, O X ) = 3 is irreducible compact hyperkähler. (Hint: use the nice multiplicative properties of

Further 7.17. The proof that X ω above is hyperkähler is done differently (more geometrically) in [DV10] . I also highly recommend the classic paper [BD85] . It turns out in both cases, the resulting hyperkähler is deformation equivalent to the Hilbert scheme of 2 points on a K3 surface.

8. Basic properties of Lagrangian fibrations of Hyperkählers, by Yajnaseni

The following exercises are based on a couple of fundamental results from [Mat99] and [Mat05] .

Given a Lagrangian fibration f : X → B of a Hyperkähler manifold X, the geometry and topology of B are heavily influenced by X. In fact, Matsushita conjectured that B P n . It is known by work of Hwang Let X be a hyperkähler manifold of dimension 2n. The following exercises show how similar the situation is in higher dimensions. The quadratic space (H 2 (X, R), q X ) controls much of the geometry of X and is a central gadget in the study of hyperkähler manifolds.

Recall that q X is a priori dependant on the symplectic form σ ∈ H 0 (X, Ω 2 X ), however, up to scaling, it is independent of σ. Here are some key properties of q X (we denote the associated bilinear form again by q X ).

• The normalized symplectic form σ satisfies q X (σ) = 0 and q X (σ + σ) = 1.

• More generally, for α i ∈ H 2 (X), we have X α 1 · · · α 2n = c X s∈Sn q X (α s(1) , α s(2) ) . . . q X (α s(2n−1) , α s(2n−2) ) for some constant c X depending only on X. As a consequence, we obtain X σσω 2n−2 = c q X (ω) n−1 .

• If a line bundle L is ample, then q X (c 1 (L)) > 0. The Kähler cone is contained in a connected component of {α ∈ H 1,1 (X, R) | q X (α) > 0}. Partial converses to these statements exist.

For instance, if L is a line bundle with q X (L) > 0 then X is projective [GHJ03, Prop. 26.13 ].

Furthermore, if q X (α) > 0 and, for every rational curve C ⊂ X, C α > 0, then α is a Kähler class [Bou01, Théorème 1.2].

• H 1,1 (X, C) is orthogonal to H 2,0 (X, C) ⊕ H 0,2 (X, C) with respect to q X .

• By [Bog96, Ver96] whenever there exists 0 = β ∈ H 2 (X, C) that satisfies q X (β) = 0, we have β n = 0 and β n+1 = 0

We begin with a Hodge index type theoerem.

Problem 8.4. Given a divisor E on X, show that if E satisfies E 2n = 0 and E · A 2n−1 = 0 for some ample bundle A, then E ∼ 0. (Hint: Use q X (tE + A) = t 2 q(E) + 2tq(E, A) + q(A) for any variable t and that (tE + A) 2n = c X q X (tE + A) n .)

Problem 8.5. Given a divisor E on X, show that if E satisfies E 2n = 0 and E · A 2n−1 > 0 for some ample line bundle A, then q X (E, A) > 0 and the following are true For the next exercise we need the definition of a Lagrangian (possibly singular) subvariety. Recall that Definition 8.8. A subvariety Y ⊂ X is said to be a Lagrangian subvariety if dim Y = 1 2 dim X and there exists a resolution of singularities µ : Y → Y such that µ * σ| Y = 0. Use the map H 2 (X, O X ) → H 0 (B, R 2 f * O X ) induced by the Leray spectral sequence and that R 2 f * O X is torsion free.)

Problem 8.11. Show that B is Q-factorial with at worst Kawamata log terminal singularities.

For the next exercise, recall and use the following where the algebra structure on the right side is given by the multiplication map.

Iitaka's C n,m conjecture then states that T X 0 to conclude that f * T B 0 Ω 1 X 0 /B 0 .)

Matsushita [Mat05] (also see [Mat99] ) extends this equality to the big open set U which includes the smooth points of the discriminant divisor D f , using Deligne's canonical extension. Then, using the reflexivity of R i f * O X and the isomorphism R n f * O X ω B , he shows that R i f * O X Ω i B .

We work through an idea of Beauville [Bea99] , following work of Yau and Zaslow [YZ96] , which uses hyperkähler geometry to count the number of rational curves in a very general K3 surface of degree 2d.

Problem 1. Assume that a K3 surface X admits an elliptic pencil -that is a map π : X → P 1 so that the general fibers are smooth genus 1 curves. Assume that all the fibers that do not have geometric genus 1 are irreducible rational curves with a single node. Count the number of rational fibers. (Hint: If R = n i=1 R i is the union of rational curves, compute the topological Euler characteristic using the formula: e(X) = e(R) + e(X \ R) and compute e(R i ).) 9.1. Hyperkählers as moduli spaces of sheaves on K3 surfaces. Let X be a very general K3 surface of degree 2d with primitive line bundle L (with L 2 = 2d) and let Π = P(H 0 (X, L)) ∼ = P d−1 .

Moduli spaces of sheaves on X are frequently hyperkähler manifolds. Here are two examples:

(1) Hilbert schemes of n points on X -denoted X [n] , this space compactifies the space of unordered distinct points on X by considering length n subschemes as their limits.

(2) Compactified Jacobians -denoted J d (X) -parametrizing coherent sheaves supported on curves C ∈ Π, which when thought of as sheaves on C are line bundles (or torsion-free sheaves of rank 1 when C is singular) of degree d.

Problem 3. Show that if X is a K3 surface, then Π contains only finitely many rational curves (curves with geometric genus 0).

Problem 4. Compute the dimension of X [n] and J d (X).

Problem 5. Show that the hyperkählers X [g] and J g (X) are birational.

There is a natural map π : J g (X) → Π which sends a coherent sheaf F to the curve in Π that it is supported on.

Problem 6. Show that the general fiber of π is an Abelian variety. Describe the fibers over a general point C ∈ Π.

Problem 7. (this is [Bea99, Prop. 2.2]) Let C be an integral curve such that the normalization C has genus ≥ 1. We show that e(J d (C)) = 0 as follows.

(1) Find a line bundle M on C which is torsion of order m (for any m > 0). (This uses the comparison between the Jacobian of C and of C.)

(2) Show that tensoring by M is a free action of Z/mZ on J d ( C).

(3) Conclude that m divides e(J d (C)) for all m > 0.

It follows by the scissor property of Euler characteristics that e(J g (X)) =

where R i ∈ Π is a rational curve and π −1 (R i ) is the fiber over R i (i.e., the set of torsion free sheaves of rank 1 and degree g supported on R i ).

Problem 8. Show that e(J g (R i )) = 1 if R i is a nodal, irreducible rational curve. (Thus by a result of Xi Chen [Che02] , if X is very general then e(J g (X)) = #{R i ∈ Π}.)

Hint: Locally at a node p ∈ R i there are only 2 types of rank 1 torsion free sheaves (1) line bundles and (2) the ideal sheaf of a point. Show that if p 1 , · · · , p g ∈ R i are the nodes then J g (R i ) is stratified into loci J g S ⊂ J g (R i ) consisting of torsion-free sheaves that are not locally free exactly at the points in a subset S ⊂ {p 1 , · · · , p g }. Conclude that the only stratum where e(J g S ) = 0 is when S = {p 1 , · · · , p g } (a single point). See also [Bea99, §3] .

It remains to actually calculate the Euler characteristic of J g (X). This relies on

(1) The birational invariance of Euler characteristic for hyperkählers (see [Huy97] or use the birational invariance of betti numbers of Calabi-Yaus [Bat00] ).

(2) The computation of the Euler characteristic of X [n] by Göttsche [Got94] (see [deC00] for a nice explanation of these results).

In particular, for a K3 surface, by (1) and (2) we have:

(# rational curves on a K3 of genus g)q g = g≥0 e(J g (X))q g = g≥0 e(X [g] )q g = Π ∞ k=1 1 1−q k e(X)

where the sum over g ≥ 0 is understood to take a very general K3 surface of genus g.

Problem 9. Compute the Euler characteristic of X [2] for any complex surface using that (1) there is a birational map h : X [2] → X (2) to the symmetric product X (2) := X 2 /Σ 2 which is given by blowing up the diagonal locus and

(2) the exceptional divisor of h is a P 1 -bundle over X.

Problem 10. Find the number of bitangents to a very general plane sextic curve C ⊂ P 2 using that a very general K3 surface of genus 2 is a double cover of P 2 branched at such a sextic.

Birational Calabi-Yau n-folds have equal Betti numbers

Variétés Kähleriennes dont la première classe de Chern est nulle

Counting rational curves on K3 surfaces

Riemannian holonomy and algebraic geometry

Holomorphic symplectic geometry: a problem list, Complex and differential geometry

La variété des droites d'une hypersurface cubique de dimension 4

Lagrangian fibrations for IHS fourfolds

On the cohomology ring of a simple hyperkähler manifold (on the results of Verbitsky)

Le cône kählérien d'une variété hyperkählérienne

Isotrivialité de certaines familles kählériennes de variétés non projectives

A simple proof that rational curves on K3 are nodal

On the Hodge and Betti numbers of hyperkähler manifolds

Un contre-exemple au théorème de Torelli pour les variétés symplectiques irréductibles

Hyper-Kähler fourfolds and Grassmann geometry

Hilbert schemes of a surface and Euler characteristics

Riemannian geometry, Mathematics: Theory & Applications

Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties

Calabi-Yau manifolds and related geometries

On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four

Base manifolds for fibrations of projective irreducible symplectic manifolds

Birational symplectic manifolds and their deformations

A global Torelli theorem for hyperkähler manifolds

Lagrangian fibrations of hyperkähler fourfolds

Minimal models and the Kodaira dimension of algebraic fiber spaces

Deformations of Lagrangian subvarieties of holomorphic symplectic manifolds

Teichmüller spaces and Torelli theorems for hyperkähler manifolds

On fibre space structures of a projective irreducible symplectic manifold

Higher direct images of dualizing sheaves of Lagrangian fibrations

Counter-example to global Torelli problem for irreducible symplectic manifolds

Desingularized moduli spaces of sheaves on a K3

A new six-dimensional irreducible symplectic variety

Lagrangian fibrations on symplectic fourfolds

On the cohomology of Kähler and hyper-Kähler manifolds

Cohomology of compact hyperkähler manifolds and its applications

Mapping class group and a global Torelli theorem for hyperkähler manifolds

Hyperkahler manifolds, Mathematical Physics (Somerville)

BPS states, string duality, and nodal curves on K3