{'fraction_non_alphanumeric': 0.05466703817219281, 'fraction_numerical': 0.013652828085817777, 'mean_word_length': 4.15393452039058, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Survey of finiteness results for hyperk\"ahler manifolds

This paper is a survey of finiteness results in hyperk\"ahler geometry. We review some classical theorems by Sullivan, Koll\'ar-Matsusaka, Huybrechts, as well as theorems in the recent literature by Charles, Sawon, and joint results of the author with Verbitsky. We also strengthen a finiteness theorem of the author. These are extended notes of the author's talk during the closing conference of the Simons Semester in the Banach Center in B\c{e}dlewo, Poland.


Introduction
In each given complex dimension 2n there are only a few known examples of compact hyperkähler manifolds up to deformation, namely, the Hilbert scheme of n points on a K3 surface S, the generalized Kummer variety K n (A) and the exceptional examples O 6 (for n = 3) and O 10 (for n = 5) given by O'Grady. A natural question to ask is if there are only finitely many compact hyperkähler manifolds in any given dimension, up to deformation. This paper surveys some of the known results in this direction.
Theorem 1.1: (Huybrechts,[Hu3]) If the second integral cohomology group H 2 (Z) and the homogeneous polynomial of degree 2n − 2 on H 2 (Z) defined by the first Pontryagin class are given, then there exist at most finitely many diffeomorphism types of compact hyperkähler manifolds of complex dimension 2n realizing this structure.
Huybrechts [Hu3] also showed that fixing the Beauville-Bogomolov-Fujiki form (i.e., giving the abelian group H 2 (Z) a ring structure) is equivalent to fixing the the first Pontryagin class.
If the diffeomorphic structure on a manifold M is given, then there are also finitely many deformation types of hyperkähler metrics on M .
Theorem 1.2: (Huybrechts,[Hu3]) Let M be a fixed compact manifold. Then there exist at most finitely many different deformation types of irreducible holomorphic symplectic complex structures on M .
Using Theorem 1.2, the author and Misha Verbitsky established the following finiteness results in [KV] about hyperkähler fibrations. Theorem 1.3: (Kamenova-Verbitsky, [KV]) Let M be a fixed compact manifold of complex dimension 2n and b 2 (M ) 7. Then there are only finitely many deformation types of hyperkähler Lagrangian fibrations M −→ CP n . François Charles has the following boundedness result for families of hyperkähler varieties up to deformation. This sharpens Kollár-Matsusaka's finiteness result for hyperkähler manifolds, because he replaces the assumption that L is ample with the weaker assumption q(L) > 0, but one must still bound the top self-intersection of L.
Theorem 1.4: (Charles, [Ch]) Let n and r be two positive integers. Then there exists a scheme S of finite type over C, and a projective morphism M −→ S such that if M is a complex hyperkähler variety of dimension 2n and L is a line bundle on M with L 2n = r and q(L) > 0, where q is the Beauville-Bogomolov form, then there exists a complex point s of S such that M s is birational to M .
For a hyperkähler manifold M , the Fujiki constant and the discriminant of the Beauville-Bogomolov-Fujiki form q are topological invariants. If we fix them instead of fixing the whole intersection form q, we can ask for finiteness given much less data than in Theorem 1.1. Here we establish the following result, which generalizes our theorem in [K] previously stated for Lagrangian fibrations.
Theorem 1.5: There are at most finitely many deformation classes of hyperkähler manifolds with a fixed Fujiki constant c and a given discriminant of the Beauville-Bogomolov-Fujiki lattice (Λ, q).
In our original proof we used a primitive vector v with q(v) = 0 coming from the line bundle L associated to the Lagrangian fibration. However, we do not necessarily need v to come from a Lagrangian fibration.
In [Saw2], Sawon proved a finiteness result for Lagrangian fibrations with several assumptions on the fibration. We'll formulate Sawon's theorem in section 4. In [K] we gave the following generalization of his result. Theorem 1.6: (Kamenova, [K]) Consider a Lagrangian fibration π : M −→ CP n such that there is a line bundle P on M with q(P ) > 0 and with a given P -degree d on the general fiber F of π, i.e., P n · F = d. Then there are at most finitely many deformation classes of hyperkähler manifolds M as above, i.e., they form a bounded family.
2 Basic results in hyperkähler geometry According to Bogomolov's decomposition theorem, [Bo], any hyperkähler manifold admits a finite covering which is a product of finitely many irreducible hyperkähler manifolds. From now on we shall assume that all hyperkähler manifolds are irreducible.
Remark 2.2: In the compact case the following two notions are equivalent: a holomorphic symplectic Kähler manifold and a manifold with a hyperkähler structure, that is, a triple of complex structures satisfying the quaternionic relations and parallel with respect to the Levi-Civita connection. This equivalence follows from Yau's solution of Calabi's conjecture ( [Bes]). Throughout this paper we assume compactness and we use the complex algebraic point of view. On H 2 (M, Z) there is a natural primitive integral quadratic form, called the Beauville-Bogomolov-Fujiki form, or BBF form for shortness. The easiest way to define it is via the Fujiki relation below. For the classical definition we refer the reader to [Bea] and [Hu2]. Remark 2.5: The form q has signature (3, b 2 − 3). It is negative definite on primitive forms and positive definite on the space Ω, Ω, ω , where Ω is the holomorphic symplectic form and ω is a Kähler form (see [V1], Theorem 6.1 and [Hu2], Corollary 23.9).
Definition 2.6: Let η ∈ H 1,1 (M ) be a real (1,1)-class on a hyperkähler manifold M . We say that η is parabolic if q(η, η) = 0. A line bundle L is called parabolic if the class c 1 (L) is parabolic.
Remark 2.7: If L is a parabolic class and P ∈ H 2 (M ) is any class, then after we substitute η = P + tL into Fujiki's formula in Theorem 2.4, and compare the coefficients of t n on both sides, we obtain 2n n P n L n = c2 n q(P, L) n .
Notice that hyperkähler manifolds have a very restricted fibration structure.
Theorem 2.8: (Matsushita,[Ma1]). Let π : M → B be a surjective holomorphic map with connected fibers from a hyperkähler manifold M to a base B, with 0 < dim B < dim M . Then dim B = 1 2 dim M , and the fibers of π are holomorphic Lagrangian (i.e., the symplectic form vanishes when restricted to the fibers).
Such a map is called a holomorphic Lagrangian fibration.
Remark 2.9: D. Matsushita ([Ma2]) proved that if the base of π is smooth and M is projective, B has the same rational cohomology as CP n . Later J.-M. Hwang ([Hw]) proved that under the same assumptions B ∼ = CP n .
Definition 2.10: A line bundle L is called semiample if L N is generated by its holomorphic sections which have no common zeros.
Remark 2.11: From semiampleness it trivially follows that L is nef, however a nef bundle is not necessarily semiample. Let π : M → B be a holomorphic Lagrangian fibration and let ω B be a Kähler class on B. Then η := π * ω B is semiample and parabolic. By Matsushita's theorem, the converse is also true: if L is semiample and parabolic, L induces a Lagrangian fibration.
Conjecture 2.12: (Hyperkähler SYZ conjecture) Let L be a parabolic nef line bundle on a hyperkähler manifold. Then L is semiample.
Remark 2.14: As a corollary to the SYZ conjecture we see that any hyperkähler manifold M with b 2 (M ) 5 can be deformed to one that admits a Lagrangian fibration. This is one of the reasons to study Lagrangian fibrations on hyperkähler manifolds. Indeed, if b 2 (M ) 5, by Meyer's theorem there is an isotropic vector v. One can deform M to a hyperkähler manifold M ′ whose Picard group is generated by a primitive line bundle L corresponding to v, i.e., q(c 1 (L)) = 0. The positive cone of M ′ coincides with the Kähler cone, and therefore, either L or L * is nef. By the SYZ conjecture, M ′ admits a Lagrangian fibration. This is the argument of Proposition 4.3 in Sawon's paper [Saw1].
Relatively recently Matsushita conjectured that the fibration structure of a hyperkähler manifold is even more restricted than the structure given by Theorem 2.16: (B. van Geemen, C. Voisin, [vGV]) Let X be a projective hyperkähler manifold of dimension 2n admitting a Lagrangian fibration f : X → B. Assume b 2,tr (X) = b 2 (X) − ρ(X) 5. Then a very general deformation (X ′ , f ′ , B ′ ) of the triple (X, f, B) satisfies Matsushita's conjecture.

Some classical results
There is a general classical result of Dennis Sullivan about finiteness of simply connected Kähler manifolds. It follows from the Q-formality of Kähler manifolds combined with the description of the rational homotopy type of Kähler manifolds in terms of a differential graded algebra. This algebra encodes the cohomology with its ring structure as a primary invariant.
Theorem 3.1: (Sullivan, [Sull]) The diffeomorphism type of a simply connected Kähler manifold (of complex dimension greater than 2) is finitely determined by its integral cohomology ring and its Pontryagin classes. Daniel Huybrechts noticed that for a hyperkähler manifold M much less of the topology needs to be fixed in order to determine the diffeomorphism type up to finite ambiguity. The first Pontryagin class p 1 (M ) ∈ H 4 (M, Z) gives rise to a homogeneous polynomial p 1 : H 2 (M, Z) −→ Z of degree 2n − 2.
Theorem 3.2: (Huybrechts,[Hu3]) If the second integral cohomology group H 2 (Z) and the homogeneous polynomial of degree 2n − 2 on H 2 (Z) defined by the first Pontryagin class are given, then there exist at most finitely many diffeomorphism types of compact hyperkähler manifolds of complex dimension 2n realizing this structure.
Here Huybrechts fixes H 2 (Z) as an abelian group. The first Pontryagin class, or equivalently, the Beauville-Bogomolov-Fujiki form, give the ring structure of the second cohomology H 2 (Z). The proof of this theorem uses Kollár-Matsusaka's finiteness result: Theorem 3.3: (Kollár-Matsusaka, [KM]) There are finitely many deformation types of projective manifolds M of dimension d that admit an ample line bundle L with fixed intersection numbers L d and K M · L d−1 ∈ Z.
In particular, for Calabi-Yau manifolds M of dimension d it is enough to fix the top intersection L d . In order to remove the fixed polarization, Huybrechts shows that any hypekähler manifold with fixed H 2 (Z) and p 1 can be deformed to one that admits a polarization L with bounded L d , using Hitchin-Sawon's formulas in [HS]. Here is a version of Huybrechts's finiteness theorem where the BBF form is given in place of p 1 .
Theorem 3.4: (Huybrechts,[Hu3]) There are only finitely many deformation types of hyperkähler manifolds M of fixed dimension such that the lattice (H 2 (M, Z), q M ) is isomorphic to a given one.
Once the diffeomorphism type of M is fixed, Huybrechts also shows that there are finitely many deformation types of hyperkähler metrics g on M .
Theorem 3.5: (Huybrechts,[Hu3]) Let M be a fixed compact manifolds. Then there exist at most finitely many deformation types of hyperkähler structures on M .
The idea of the proof is the following. Let Λ = (H 2 (M, Z), q) and M Λ be the coarse moduli space of marked hyperkähler manifolds (M, ϕ). Fix a primitive positive element v ∈ H 2 (M, Z), i.e., q(v) > 0. An isomorphism, or a marking, ϕ : (H 2 (M, Z), q) −→ Λ determines a point (M, ϕ) ∈ M Λ . By Huybrechts's proectivity criterion [Hu1], since q(v) > 0, M is deformation equivalent to a projective manifold M ′ with an ample line bundle L corresponding to v. By Fujiki's formula in Theorem 2.4, L 2n = c · q(L) n = c · q(v) n . Fujiki's constant c depends only on the topological type of M , which is fixed. Since v is also fixed, q(v) is determined, and therefore the top intersection L 2n is known and one applies Kollár-Matsusaka's Theorem 3.3.

Some recent results
Together with Misha Verbitsky in [KV] we were interested in studying finiteness questions about hyperkähler manifolds that admit Lagrangian fibrations. As we pointed out in Remark 2.14, if the SYZ conjecture holds, then any hyperkähler manifold (with b 2 5) can be deformed to one that admits a Lagrangian fibration, and therefore it would be enough to study Lagrangian fibrations for questions concerning deformation classes of hyperkähler varieties. By Huybrechts's Theorem 3.5, for a fixed compact manifold there are at most finitely many deformation types of hyperkähler structures on it. We proved that for each hyperkähler structure there are only finitely many ways in which it fibers over a smooth base. where L remains of type (1, 1) on deformations of the complex structure. In [KV] we proved that there are finitely many orbits of the action of Γ I on Teich L using lattice theory methods and Nikulin-style techniques applied to the BBF form. If we take a general deformation of a Lagrangian fibration, the Picard group would be 1-dimensional and generated by the nef line bundle L. For each such pair (M, L) we prove that there is a unique deformation type of a fibration structure. Since there are finitely many orbits of Γ I acting on Teich L we conclude finiteness of the deformation types of Lagrangian fibrations.
Recently, François Charles sharpened Kollár-Matsusaka's theorem for hyperkähler varieties by replacing the ampleness assumption with a weaker one. Our results, Theorem 4.4 and Theorem 4.9, rely on his theorem. Consider a lattice Λ, i.e., a free Z−module of finite rank together with a non-degenerate symmetric bilinear from q with values in Z. If {e i } is a basis of Λ, the discriminant of Λ is discr(Λ) = det(q(e i , e j )). Let us recall the following lemma from [K]. Lemma 4.3: Let (Λ, q) be an indefinite lattice and v ∈ Λ be an isotropic nonzero vector. Then there exists a positive vector w ∈ Λ such that 0 < q(w, v) |discr(Λ)| and 0 < q(w, w) 2|discr(Λ)|.
One of our main results in [K] was the following finiteness theorem with the assumption of existence of a Lagrangian fibration. Here we have dropped that assumption and we obtain a more general result. Proof: If b 2 (M ) = 3 or 4, there are finitely many lattices of this rank with fixed discriminant (see Theorem 1.1, chapter 9 of [Cas]). For each one of these finitely many lattices the positive vector w with minimal positive intersection q(w, w) has bounded self-intersection and we apply Charles's Theorem 4.2 directly. Now assume that b 2 5. By Meyer's theorem on indefinite lattices of rank at least 5, there exists a non-trivial isotropic vector v ( [Cas]). We apply Lemma 4.3 for (Λ, q) = (H 2 (X, Z), q) and v = v. Then there exists a positive vector w such that 0 < q(w, w) 2|discr(Λ)|. By Fujiki's formula, 0 < w 2n = c · q(w, w) n c · (2|discr(Λ)|) n , i.e., w 2n is bounded by the given invariants. Deform M to a hyperkähler manifold M ′ with a line bundle L corresponding to the class w. For each top intersection c 1 (L) 2n in the interval (0, (2|discr(Λ)|) n ], we obtain only finitely many deformation classes of M by Charles's Theorem 4.2.
Since the families of hyperkähler manifolds above form a bounded family, there are only finitely many choices of the second Betti number. The lattice (H 2 (M, Z), q) encodes substantial information for hyperkähler manifolds and b 2 (M ) is an important invariant, therefore we list separately the following direct corollary.
Corollary 4.5: In the assumptions of Theorem 4.4, i.e., given numbers n ∈ Z + , c ∈ Q and discriminant d ∈ Z, the second Betti number b 2 (M ) is bounded.
We would like to mention also the following theorem in the recent literature.
Theorem 4.6: (Sawon,[Saw2]) Fix positive integers n and d 1 , · · · , d n , with d 1 |d 2 | · · · |d n . Consider Lagrangian fibrations π : M → CP n that satisfy: (1) π : M → CP n admits a global section; (2) there is a very ample line bundle on M which gives a polarization of type (d 1 , · · · , d n ) when restricted to a generic smooth fiber M t ; (3) over a generic point t of the discriminant locus the fiber M t is a rank-one semi-stable degeneration of abelian varieties; (4) a neighbourhood U of a generic point t ∈ CP n describes a maximal variation of abelian varieties.
Then there are finitely many such Lagrangian fibrations up to deformation.
Justin Sawon's proof is based on the following observations. The existence of a section implies that there is a distinguished point in each fiber. Together with the given type of a polarization it gives a natural classifying map ϕ : CP n \ ∆ −→ A d1,··· ,dn , where ∆ is the discriminant locus of π. Sawon extends ϕ to ϕ : CP n \ ∆ 0 −→ A * d1,··· ,dn , where A * d1,··· ,dn is a partial compactification of A d1,··· ,dn and ∆ 0 ⊂ ∆ is of codimension 2 in CP n . He chooses an ample line bundle H on A * d1,··· ,dn and bounds deg(ϕ * H), which is non-zero by (4). Thus, ϕ belongs to finitely many families of rational maps CP n A * d1,··· ,dn , and finiteness of Lagrangian fibrations as above follows.
Remark 4.7: Notice that as a corollary of Matsushita's conjecture, part (4) of Sawon's Theorem simply excludes locally isotrivial fibrations. We need to apply only the deformational version (see van Geemen-Voisin's Theorem 2.16) of Matsushita's conjecture to Sawon's theorem in order to remove the seemingly restrictive assumption (4).
Remark 4.8: If there is a section σ : CP n −→ M , this means that σ(CP n ) is a Lagrangian subvariety in M . Finding Lagrangian subvarieties in a hyperkähler manifold is itself a very interesting problem (for example, see [HT]).
Using the methods above combined with F. Charles' Theorem 4.2, in [K] we generalized Sawon's Theorem 4.6.
Theorem 4.9: Consider a Lagrangian fibration π : M −→ CP n such that there is a line bundle P on M with q(P ) > 0 and with a given P -degree d on the general fiber F of π, i.e., P n · F = d. Then there are at most finitely many deformation classes of hyperkähler manifolds M as above, i.e., they form a bounded family.