FINITENESS OF POLARIZED K3 SURFACES AND HYPERKÄHLER MANIFOLDS

. — In the moduli space of polarized varieties ( X, L ) the same unpolarized variety X can occur more than once. However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the ‘orbit’ of X , i.e. the subset { ( X i , L i ) | X i ’ X } , is known to be ﬁnite, which may be viewed as a consequence of the Kawamata–Morrison cone conjecture. In this note we provide a proof of this ﬁniteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the ﬁniteness by means of Baily–Borel type arguments. We also address related questions concerning ﬁniteness in twistor families associated with polarized K3 surfaces of CM type. Résumé. —

The paper studies the connection between moduli spaces of polarized varieties, on the one hand, and the shape of the ample cone on a fixed variety, on the other hand.To illustrate our point of departure, let us revue a few well-known results.0.1.The classical Torelli theorem shows that two complex smooth projective curves C and C are isomorphic if and only if their polarized Hodge structures are isomorphic, i.e. there exists a Hodge isometry H 1 (C, Z) H 1 (C, Z), or, equivalently, if their principally polarized Jacobians J(C) J(C ) are isomorphic.Dropping the compatibility with the polarizations, so only requiring isomorphisms of unpolarized Hodge structures or unpolarized abelian varieties, the geometric relation between C and C becomes less clear.In moduli theoretic terms, one may wonder about the geometric nature of the quotient map M g → M g / ∼ .Here, M g denotes the moduli space of genus g curves and C ∼ C if and only if J(C) J(C ) unpolarized.
Similarly, two polarized K3 surfaces (S, L) and (S , L ) are isomorphic if and only if there exists a Hodge isometry H 2 (S, Z) H 2 (S , Z) that maps L to L .Dropping the latter condition has a clear geometric meaning and corresponds to considering isomorphisms between unpolarized surfaces S and S .Therefore, dividing out by the resulting equivalence relation yields a map M d → M d / ∼ from the moduli space of polarized K3 surfaces (S, L) of degree d to the space of isomorphism classes of K3 surfaces that merely admit a polarization of this degree.Considering only isomorphisms of Hodge structures without any further compatibilities leads to the analogue of the aforementioned question for curves.At this time, there is no clear picture of what the existence of an unpolarized isomorphism of Hodge structures could mean for the geometry of the two K3 surfaces, but finiteness has recently been established in [Efi17].
Other types of varieties, like abelian varieties, Calabi-Yau or hyperkähler varieties, can be discussed from the same perspective.0.2.Let us move to the cone side.For a K3 surface S, the ample cone Amp(S) ⊂ NS(S) ⊗ R and its closure, the nef cone Nef(S), are complicated and usually not rationally polyhedral.The situation changes when the natural action of Aut(S) is taken into account.More precisely, there exists a fundamental domain Π ⊂ Nef + (S) for the action of Aut(S) on the effective nef cone that is rational polyhedral, see [Ste85] or [Huy16,Chapter 8].Its generalization to smooth projective varieties with trivial canonical bundle is the cone conjecture of Kawamata [Kaw97] and Morrison [Mor93].It has been proved for abelian varieties [PS12] and for hyperkähler manifolds [AV17,AV16,MY15].For recent progress in the case of Calabi-Yau varieties see [LOP15].
The cone conjecture has the following somewhat less technical consequence: Up to the action of the group of automorphisms, there exist at most finitely many polarizations of a fixed degree, see [Ste85] for the argument.For abelian varieties the result had been observed already in [NN81].0.3.These two circles of ideas are of course linked.For example, for K3 surfaces the fibre over S 0 of the map M d → M d / ∼ is naturally identified with the set {L | ample, (L) 2 = d}/ Aut(S 0 ).Due to the cone conjecture for K3 surfaces, this set is finite and, therefore, all fibres of M d → M d / ∼ are finite.Similarly, for the moduli space A g,d of polarized abelian varieties of dimension g and degree d and a fixed (unpolarized) abelian variety A 0 there exist at most finitely many polarized abelian varieties (A 1 , L 1 ), . . ., (A n , L n ) ∈ A g,d with A i A 0 , i.e. the fibres of A g,d → A g,d / ∼ , and consequently also of M g → M g / ∼ , are finite.
It is worth emphasizing that the quotients M d / ∼ , M g / ∼ , and A g,d / ∼ have no reasonable geometric structure, which is mainly due to the fact that the fibres of the quotient maps are all finite but of unbounded cardinality.See Section 1.5, where this is discussed for K3 surfaces.
Note that the local Torelli theorem for these types of varieties immediately implies that the fibres of and are discrete subsets of the corresponding moduli spaces.The present paper is motivated by the question whether the geometric nature of the three moduli spaces M d , A g,d , and M g (and others), namely being quasi-projective varieties, can alternatively be used to deduce from their discreteness the finiteness of the three sets M d (S 0 ), A g,d (A 0 ), and M g (C 0 ).There are well-known instances where this naive idea indeed yields finiteness of certain discrete sets in appropriate moduli spaces by verifying their algebraicity.As an example, we recall in Section 1.6 the proof for the finiteness of Aut(S, L) along these lines.
Although, finiteness of the sets M d (S 0 ), A g,d (A 0 ), or M g (C 0 ) cannot be deduced quite so easily, we will show that the quasi-projectivity of certain related moduli spaces can indeed be exploited.We will demonstrate this for the moduli space M d of compact hyperkähler manifolds of fixed degree and fixed dimension by proving the following result.
In other words, the set of ample line bundles on X 0 of fixed degree is finite up to the action of the group Aut(X 0 ) of automorphisms of X 0 .As the cone conjecture for hyperkähler manifolds has recently been established in great generality in [AV17,AV16], see also [MY15] for a proof for the two standard series, the theorem can also be seen as a consequence of the cone conjecture.In fact, the full conjecture is not needed to conclude the above result from the global Torelli theorem, a shortcut is outlined in Section 1.4.However, our approach shows that finiteness results of this type can be deduced more directly and without using any version of the global Torelli theorem from moduli space considerations and Baily-Borel type arguments.Alternatively, Griffiths' extension theorem can be used.To complete the picture, we shall outline in Section 3 a proof of Theorem 0.1 that reduces the assertion to the case of abelian varieties via the Kuga-Satake construction.0.4.Finiteness results of this type are clearly fundamental.However, as they fail in the non-algebraic setting, they are also quite remarkable.Recall that there indeed exist non-isotrivial families of K3 surfaces or hyperkähler manifolds with a dense subset of fibres all isomorphic to one of the fibres.More precisely, the set of period points corresponding to K3 surfaces or hyperkähler manifolds isomorphic to a fixed one is dense except when the complex plane (H 2,0 ⊕ H 0,2 ) contains a non-trivial integral class, see [Ver15,Ver17] and [BL16,Remark 5.7].Theorem 0.1 now says that this cannot happen for polarized families.The second goal of this paper is to prove a similar result for certain families 'orthogonal' to the polarized case.More precisely, we study families provided by the twistor space construction.Let us restrict to K3 surfaces for simplicity and recall that associated with any K3 surface S 0 endowed with a Kähler class ω 0 , e.g. the one given by a polarization L 0 , one associates a twistor family S → P 1 of K3 surfaces S t , t ∈ P 1 , with a natural Kähler class ω t .Only countably many of the fibres S t are projective and the complex manifold S is not even Kähler.We then prove the following non-algebraic analogue of Theorem 0.1, see Proposition 2.8.Theorem 0.2.-Let S → P 1 be the twistor space associated with a polarized K3 surface (S 0 , L 0 ).Assume that S 0 has CM.Then at most finitely many twistor fibres S t are isomorphic to S 0 .
Despite the similarities between the two finiteness results, they are rather different from another perspective.Namely, in Theorem 0.1 the number |M d (S 0 )| is finite but unbounded for varying S 0 , whereas in Theorem 0.2 the number of fibres isomorphic to S 0 only depends on the CM field and can be universally bounded by 132, cf.Remark 2.9.At this point, it is not clear whether the assumption on S 0 to have CM is really necessary, but the proof suggests that it might.0.5.We are also interested in the metric aspect.Recall that to any polarization L on a compact hyperkähler manifold X there is naturally associated a hyperkähler metric g L on the underlying manifold.The resulting Riemannian manifold shall be denoted (X, g L ).From the Riemannian perspective it is then natural to wonder how often the same Riemannian manifold occurs for (X, L) ∈ M d .Again restricting to the case of K3 surfaces for simplicity, we prove the following result, see Corollary 2.3.
for which there exists an isometry (S 0 , g L 0 ) (S, g L ) of the underlying Riemannian manifolds is finite.

Finiteness of polarizations on hyperkähler manifolds
Without using the global Torelli theorem, finiteness of polarizations of fixed degree on a compact hyperkähler manifold can be proved by applying Baily-Borel type arguments that ensure that certain arithmetic quotients and maps between them are algebraic.
Remark 1.1.-We shall use the following elementary facts from lattice theory, cf.[Kne02, Satz 30.2], the second being a special case of the first.Let N and Λ be arbitrary lattices.
(1) Up to the action of the orthogonal group O(Λ), there exist at most finitely many (primitive) embeddings η i : N → Λ, i = 1, . . ., k.In the following, we often denote the lattices given as the orthogonal complements by (2) Up to the action of O(N ), there exist only finitely many (primitive) classes ∈ N with fixed square ( ) 2 = d.(3) If Λ is definite, then there exist at most finitely many embeddings N → Λ.

1.2.
Assume now that N has signature (1, ρ(X) − 1) and fix an element ∈ N with ( ) 2 = d.Then consider the moduli stack M (N, ) of (N, )-polarized hyperkähler manifolds of deformation type (or just diffeomorphic or homeomorphic to) X 0 .So, M (N, ) (T ) consists of families π : X → T of compact hyperkähler manifolds deformation equivalent to X 0 together with an embedding ι : N → R 2 π * Z of locally constant systems on T which fibrewise induces a primitive embedding of lattices N → Pic(X t ) NS(X t ) H 1,1 (X t , Z) ⊂ H 2 (X t , Z) mapping to an ample line bundle L t on X t .
Standard moduli space constructions yield the following result.
Proposition 1.2.-The stack M (N, ) of (N, )-polarized compact hyperkähler manifolds deformation equivalent to X 0 is a Deligne-Mumford stack with a quasipolarized coarse moduli space M (N, ) .
Proof.-We sketch the main steps in the construction.Variants of this can be found in the literature, see [Bea04,Dol96] and Section 4 for further comments.
Denote by M d the moduli stack of polarized hyperkähler manifolds (X, L) of degree d.This is a Deligne-Mumford stack with a quasi-projective coarse moduli space, see [Vie95] and [Huy16, Chapter 5] for further references in the case of K3 surfaces. (1)The map (X → T, ι) → (X → T, ι( )) defines a morphism f : M (N, ) → (1) The degree of a polarization L would usually be given as the top intersection form (L) 2m or, equivalently, as c F •q(L) m with the Fujiki constant c F .As we are only looking at varieties deformation M d .As there exist at most finitely many isometric embeddings ⊥ → ι( ) ⊥ ⊂ Pic(X) (use Remark 1.1 (3)), the morphism is quasi-finite.In [Bea04] M (N, ) is realized as an open and closed substack of Pic ρ(X) It is not difficult to show that f : M (N, ) → M d is actually proper and hence finite.Therefore, also the induced morphism between their coarse moduli spaces M (N, ) → M d is finite.For the existence of the coarse moduli spaces (as algebraic spaces) one needs to use the finiteness of the stabilizers (Matsusaka-Mumford, as for K3 surfaces), see [KM97].Using that M d is quasi-projective [Vie95], yields the quasi-projectivity of M (N, ) .
According to Remark 1.1, there exist, up to the action of O(N ), at most finitely many 1 , . . ., m ∈ N with ( j ) 2 = d.In this sense, the quasi-projective variety can be understood as the moduli space of N -polarized hyperkähler manifolds of deformation type X 0 and degree d. (2)  As explained in the above proof, mapping (X → T, ι : to the moduli stack M d of polarized compact hyperkähler manifolds of deformation type X 0 and degree d.This yields a morphism between their quasi-projective coarse moduli spaces Its image consists of all the points that correspond to polarized hyperkähler manifolds (X, L) of deformation type X 0 for which the polarization L is contained in a primitive sublattice of the Néron-Severi lattice abstractly isomorphic to N .In particular, for equivalent to X 0 , the Fujiki constant is fixed and so prescribing the Beauville-Bogomolov square q(L) or the classical degree (L) 2m amounts to the same.In particular, it would be enough to fix the topological type of X 0 in our discussion.
(2) . . .with the slight ambiguity that a primitive embedding of N may send more than one j to an ample class.
(3) As a side remark, but very much in the spirit of our discussion, the fact that M N,d is quasiprojective allows one to circumvent Remark 1.1 (1), as it implies that there can be at most finitely many equivalence classes of embeddings N → Λ obtained as composition Remark 1.3.-The image of M i N, j → M d defines a special cycle in the sense of [Kud13].In particular, each M d (X 0 ) is contained in a finite union of special cycles of codimension ρ(X 0 ) − 1.

1.3.
For each of the lattices T i = η i (N ) ⊥ ⊂ Λ we consider the usual period domain defined by the closed condition (x) 2 = 0 and the open condition (x.x) > 0. Recall that D i can be identified with the Grassmannian of positive oriented planes in T i ⊗ R and that it consists of two connected components, cf.[Huy16,Chapter 6].
The orthogonal group O(T i ) acts naturally on D i and due to Baily-Borel [BB66] the quotient O(T i ) \ D i is a quasi-projective variety with finite quotient singularities.
Proposition 1.4.-Mapping (X, ι : N → NS(X)) to its period ϕ(H 2,0 (X)) yields a well defined and algebraic map Proof.-Note that the primitive embedding ι : N → NS(X) composed with the inclusion NS(X) H 1,1 (X, Z) ⊂ H 2 (X, Z) and a marking H 2 (X, Z) Λ yields a primitive embedding η : N → Λ.By definition of M i N,d this primitive embedding is equivalent to η i and, hence, there indeed exists a marking ϕ with ϕ • ι = η i .In particular, the sublattices T i = η i (N ) ⊥ and ϕ(ι(N )) ⊥ of Λ coincide.Hence, ϕ(H 2,0 (X)) ⊂ T i ⊗ C, which thus defines a point in D i .Changing ϕ to ϕ still satisfying ϕ • ι = η i yields a period in the same orbit of the O(T i )-action on D i .It is easy to check that isomorphic N -polarized (X, ι) and (X , ι ), i.e. both defining the same point in M i N,d , yield the same point in O(T i ) \ D i .Introducing markings globally over M i N,d (by passing to the appropriate principal bundle) and applying local period maps, one finds that π i : The crucial input now is Borel's result [Bor72] which shows that π i is automatically algebraic.Note that Borel's result only allows quotients by torsion free groups.However, introducing finite level structures one obtains a finite cover of M i N,d that then maps holomorphically to a smooth quotient Γ \ D i for some finite index torsion free subgroup Γ ⊂ O(T i ).
-By the local Torelli theorem, non-trivial local deformations of (X, η) are detected by their periods, i.e. by their images under π i .Thus, the fibres of π i are discrete.Now use that π i is algebraic, which immediately implies finiteness.
1.4.Proofs of Theorem 0.1.As announced in the introduction, we shall now present a proof of the finiteness of M d (X 0 ) for hyperkähler manifolds X 0 that avoids the Kawamata-Morrison cone conjecture and, in fact, the global Torelli theorem.We shall also sketch an argument that uses the global Torelli theorem directly.For a third proof via the Kuga-Satake construction see Section 3.
-Let X 0 be a compact hyperkähler manifold which is assumed to be projective.Hence, N Λ is a primitive sublattice of signature (1, ρ(X 0 ) − 1) (which by [Huy99] is known to be equivalent to the projectivity of X 0 ).
The set M d (X 0 ) ⊂ M d of all polarized (X, L) of degree d with X X 0 is contained in the image of the map (1.1).So, let (X, ι) ∈ M N,d with X X 0 .Then ι : N ∼ → NS(X), because an embedding of abstractly isomorphic lattices is automatically an isomorphism.Hence, (X, ι) and (X 0 , ι 0 : N → NS(X) ⊥ , which shows that (X, ι) and (X 0 , ι 0 ) have the same image under π i : ) is contained in the image under the map (1.1) of a fibre of π i .Now use Corollary 1.5, to conclude the finiteness of M d (X 0 ).
Remark 1.6.-For Calabi-Yau threefolds a similar idea has been exploited by Szendrői [Sze99] and it seems plausible that his arguments can be generalized to cover Calabi-Yau manifolds of arbitrary dimensions.Instead of Baily-Borel, so the algebraicity of the holomorphic period map, he applies Griffiths' extension theorem [CGGH83] which ensures the existence of a proper holomorphic extension.
In fact, quite generally, quasi-finiteness can alternatively be deduced from the extension theorem.See [JL18] for an in depth discussion of various aspects of the quasi-finiteness of period maps and references.
-As has been mentioned, Theorem 0.1 can also be deduced from the global Torelli theorem which in turn relies on the existence of Ricci-flat metrics, twistor spaces, etc.Here is a quick outline of the argument.The reader may, for simplicity, restrict to the case of K3 surfaces.
The group of diffeomorphisms acts by a finite index subgroup on H 2 (X 0 , Z), often called the monodromy group Mon(X 0 ) ⊂ O(H 2 (X 0 , Z)).For K3 surfaces the index has been determined in [Bor86] (finiteness was known to Weil).The same argument combined with [Huy03] proves finiteness for arbitrary hyperkähler manifolds.The subgroup respecting the Hodge structure yields a finite index subgroup Mon Hdg (X 0 ) ⊂ O(NS(X 0 )).(The finite index of the inclusion NS(X 0 ) ⊕ NS(X 0 ) ⊥ ⊂ H 2 (X 0 , Z) intervenes here.)We know that O(NS(X 0 )) acts with finitely many orbits on the set of all ∈ NS(X 0 ) with fixed ( ) 2 = d, see Remark 1.1.The last step now is to describe the image of Aut(X 0 ) → O(NS(X 0 )) as the subgroup of finite index of all g ∈ O(NS(X 0 )) such that g maps (at least) one Kähler class again to a Kähler class.This last assertion is part of the global Torelli theorem, cf.[Huy16, Chapter 15] for the case of K3 surfaces and [Mar11,Ver13] for the higher-dimensional case.
Corollary 1.7.-Let π : (X , L) → T be a polarized family of compact hyperkähler manifolds (e.g. of polarized K3 surfaces) over a quasi-projective base T .Then for any X 0 the set {t ∈ T | X t X 0 } is Zariski closed.
Corollary 1.8.-Let N be a lattice of signature (1, m) and ∈ N a primitive class.Then for any compact hyperkähler manifold X 0 there exist, up to the action of Aut(X 0 ), at most finitely many isometric embeddings N → NS(X 0 ) mapping to an ample class.
Proof.-The result can be proved in the spirit of the discussion above, by means of moduli spaces of lattice polarized hyperkähler manifolds and again using Baily-Borel arguments.It can also be deduced from Theorem 0.1 directly.Indeed, up to the action of Aut(X 0 ) there are only finitely many ample classes L ∈ NS(X 0 ) with (L) 2 = ( ) 2 .An isometric embedding N → NS(X 0 ) mapping to a fixed L is then determined by the induced ⊥ → L ⊥ .As L ⊥ is negative definite by the Hodge index theorem, there are only finitely many such embeddings, see Remark 1.1.

1.5.
The cardinality of the 'orbits' cannot be bounded.For abelian surfaces this has been observed in [Hay68,Lan06].This suggests a similar behavior for the associated Kummer surfaces, but controlling the degree of the polarizations is technical.Here, we exhibit an example of polarized K3 surfaces (S, L) of degree (L) 2 = d = 4d 0 for any fixed odd d 0 > 0 not divisible by any p 3 showing that |M d (S)| cannot be bounded.
For this, consider an increasing sequence of primes p i ≡ 3 (4).By Siegel's theorem, the sequence of class numbers ).It is classically known that two forms are in the same genus if they differ by forms in Cl(K i ) 2 (Gauss principal genus theorem) and that under our assumptions in fact Cl(K i ) 2 = Cl(K i ).cf. [Cox89, Proposition 3.11].Hence, for all i there exist non-isomorphic Γ i1 , . . ., Γ ih i within the same genus.
Hence, the indefinite lattices Γij := Γ ij ⊕ Z(−d 0 ), j = 1, . . ., h i , of rank three are in the same genus for fixed i.However, the genus of an indefinite ternary form determines the isomorphism class assuming that its discriminant is odd and indivisible by any cube, cf.[CS98, Chapter 15, Theorem 21].Hence, Γi1 • • • Γih i and we shall denote this isomorphism type by Γi .Then the generators of Z(−d 0 ) correspond to elements α ij ∈ Γi .As their orthogonal complements α ⊥ ij are isomorphic to Γ ij , the orbits O( Γi ) • α ij ⊂ Γi are all distinct.Finally, set N i := Γi (−4), which is lattice of signature (1, 2) containing classes α i1 , . . ., α ih i of square (α ij ) 2 = d = 4d 0 with distinct O(N i )-orbits.By the surjectivity of the period map, there exist K3 surfaces S i with NS(S i ) N i .As the lattices N i do not contain any (−2)-classes, up to sign the α ij , j = 1, . . ., h i correspond to ample line bundles L ij with pairwise distinct O(NS(S i ))-orbits and, hence, pairwise distinct Aut(S i )-orbits.In other words, 1.6.We conclude this section by a few additional remarks.
-The kind of finiteness we have discussed for K3 surfaces, hyperkähler manifolds, and abelian varieties does not generalize to arbitrary varieties.In fact, it already fails for blow-ups of K3 surfaces.where E → ∆ is the exceptional divisor.Then for an infinite orbit {s i := f i (s)} the infinitely many fibres X s i are all isomorphic.These isomorphisms will be mostly unpolarized, as {f i * L} will be infinite for every ample line bundle L on S. Hence, in the corresponding moduli space, the orbit M d−1 (Bl s (S)) will be infinite.
-There are indeed instances where finiteness can be deduced by combining a local argument, showing discreteness of a set, with algebraicity.For example, for a polarized K3 surface (S, L) the quasi-projectivity of the Hilbert scheme Hilb P (S × S) (of closed subschemes of S × S with fixed Hilbert polynomial P (n) = χ(S, L 2n )) implies that the group Aut(S, L) of automorphisms f : S ∼ → S with f * L L is finite, cf.[Huy16, Chapter 5].Note that the finiteness of the group of polarized automorphisms implies that the moduli stacks discussed previously are Deligne-Mumford stacks.

Finiteness of hyperkähler metrics and in twistor families
Let X be a compact hyperkähler manifold, for example a K3 surface S, and ω ∈ H 1,1 (X) a Kähler class.Then there exists a unique hyperkähler metric g on X whose Kähler form g(I , ) represents ω.This can in particular be applied to the Kähler class provided by the first Chern class of an ample line bundle L on X.The associated hyperkähler metric shall be denoted g L and then (X, g L ) is the underlying Riemannian manifold (with the complex structure of X dropped).
We shall first address the question how many polarized hyperkähler manifolds (X, L) ∈ M d of fixed degree realize the same Riemannian manifold, i.e. for a given (X 0 , L 0 ) we study the set M d (X 0 , g L 0 ) ⊂ M d of all (X, L) ∈ M d such that there exists an isometry (X 0 , g L 0 ) (X, g L ) between the underlying Riemannian manifolds.We will then turn to families 'orthogonal' to the moduli spaces M d provided by the twistor construction.Recall that to each hyperkähler metric g there exists an S 2 of complex structures compatible with g.This leads to the twistor family π : X → P 1 consisting of a complex manifold X with underlying differentiable manifold X × P 1 and the holomorphic projection π to the second factor.Each fibre X t , t ∈ P 1 , comes with an associated Kähler class ω t ∈ H 1,1 (X t ).Altogether they span a positive three-space ω t t∈P 1 ⊂ H 2 (X, R), which is alternatively described as Here, 0 = σ ∈ H 2,0 (X) is the unique (up to scaling) holomorphic two-form.We shall denote by X 0 the fibre that corresponds to X and so ω 0 = ω.Note that the Beauville-Bogomolov form is constant on {ω t } or, in other words, ω 2n t ≡ const.We then ask whether there are fibres X t , t ∈ P 1 , biholomorphic to X or such that (X t , ω t ) (X, ω) as Kähler manifolds?Are the following sets finite: We prove finiteness in the first case and for K3 surfaces with CM in the second.

2.1.
Consider the twistor family X → P 1 associated with a compact hyperkähler manifold X endowed with a Kähler class.
Lemma 2.1.-There exist only finitely many t ∈ P 1 such that the natural Kähler class ω t is integral, i.e. given by an ample line bundle L t on X t .In particular, if ω is the class of an ample line bundle L, then at most finitely many polarized fibres (X t , ω t ) are isomorphic to (X, L).
Proof.-The positive three-space ω t = (H 2,0 ⊕ H 0,2 )(X, R) ⊕ R • ω intersects the lattice H 2 (X, Z) in a positive definite lattice (of rank 3).As the number of classes of fixed square in a positive definite lattice is finite, the set {ω t } ∩ H 2 (X, Z) is finite.As the class ω t determines t ∈ P 1 up to complex conjugation, for only finitely many fibres X t the class ω t can be integral.
Remark 2.2.-Note that in most cases the natural class ω t on X t will be integral or just rational for only two of the fibres.Indeed, ω t ∈ P := R•Re(σ)⊕R•Im(σ)⊕R•ω and P ∩ H 2 (X, Q) = Q • ω for very general (X, ω := c 1 (L)) and then only the fibres corresponding to X and its complex conjugate (corresponding to the positive plane (H 2,0 ⊕ H 0,2 )(X, R) with reversed orientation) have rational ω t .At the other extreme, P is defined over Q if and only if ρ(X) = h 1,1 (X).
The remaining cases with P ∩ H 2 (X, Q) of dimension two are parametrized by a countable union of real Lagrangians in Let us rephrase Lemma 2.1 more algebraically in the case of K3 surfaces.Consider the moduli space M d of polarized K3 surfaces (S, L) of degree d.For (S 0 , L 0 ) ∈ M d let M d (S 0 , g L 0 ) ⊂ M d be the set of polarized K3 surfaces (S, L) such that the underlying Riemannian manifold (S, g L ) is isometric to (S 0 , g L 0 ).
Corollary 2.3.-The set M d (S 0 , g L 0 ) ⊂ M d is always finite.For a very general (S 0 .L 0 ) the set M d (S 0 , g L 0 ) consists of (S 0 , L 0 ) and its conjugate ( S0 , L * o ).However, there exist polarized K3 surfaces Proof.-Indeed, (S, L) ∈ M d (S 0 , g L 0 ) if and only if S and S 0 with the Kähler classes induced by L and L 0 are isomorphic to fibres (S t , ω t ) and (S 0 , ω 0 ) of one twistor family S → P 1 .However, as explained above, for only finitely many fibres of the twistor family associated with (S 0 , ω 0 ) the Kähler class ω t can be integral.In fact, as discussed in Remark 2.2, for (S 0 , L 0 ) in the complement of the countable union L α ⊂ M d of real Lagrangians, only for the fibres S t corresponding to S 0 and to its conjugate S0 the class ω t will be integral and thus correspond to the Kähler class of the form g L .
To construct examples with more interesting M d (S 0 , g L 0 ) ⊂ M d , take (S 0 , L 0 ) with (L 0 ) 2 = 2, T (S 0 ) = ( 2 1 1 2 ), and such that T (S 0 ) ⊕ Z • L ⊂ H 2 (S 0 , Z) is primitive.Then the fibres (S t , ω t ) of the form (S, L) ∈ M 2 correspond to elements α ∈ T (S 0 ) ⊕ Z(2) with (α) 2 = 2.For example, S 0 corresponds to the basis vector e 3 and another fibre S t corresponds to the standard vector e 1 .As their orthogonal complements T (S 0 ) and T (S t ) Z • (e 1 − 2e 2 ) ⊕ Z • e 3 have distinct discriminants, the two fibres are not isomorphic or complex conjugate to each other.Hence, in this case It should be possible, using the above construction, to show that |M d (S 0 , g L 0 )| is unbounded for varying (S 0 , L 0 ) ∈ M d and fixed d, analogously to Section 1.5.
Remark 2.4.-The two sets M d (S 0 ), M d (S 0 , g L 0 ) ⊂ M d associated with a polarized K3 surface (S 0 , L 0 ) ∈ M d are not related and, in particular, not contained in each other.Indeed, |M d (S 0 , g L 0 )| 2 for all K3 surfaces S 0 with (H 2,0 ⊕ H 0,2 )(S 0 , Q) = 0.However, the latter condition is unrelated to the question how many non-isomorphic polarizations L on S 0 there are with (L) 2 = (L 0 ) 2 .Also, the M d (S 0 ) come in algebraic families parametrized by quasi-projective varieties, see Corollary 4.1, whereas the M d (S 0 , g L 0 ) come in families parametrized by the Lagrangians L α discussed above.

2.2.
The lattice theory in the unpolarized situation is more involved and we will restrict for simplicity again to the case of K3 surfaces.
Let T be a lattice of signature (2, n − 2) with a fixed basis γ 1 , . . ., γ n ∈ T .We consider Hodge structures of K3 type on T .Up to scaling, such a Hodge structure is given by a class σ ∈ T ⊗ C with (σ) 2 = 0 and (σ.σ) > 0. We let σ i := (γ i .σ),where we may choose σ such that σ 1 = 1 (after permuting the γ i if necessary or by assuming (γ 1 ) 2 > 0 from the start).The period field of σ is defined as which is also generated by the coordinates of σ.The Hodge structure determined by σ is general if there does not exist a proper primitive sublattice T ⊂ T with σ ∈ T ⊗ C.
Consider now an isometric embedding ϕ : Note that then λ is automatically an algebraic integer, for it is an eigenvalue of the composition Lemma 2.5.-Assume that σ ∈ T ⊗ C defines a general Hodge structure of K3 type on T such that K σ is a subfield of a CM field.Then there exist at most finitely many isometric embeddings ϕ : T → T ⊕ Z • e satisfying (2.1).
From the finiteness of the λ's one concludes the assertion by observing that λ determines ϕ essentially uniquely.Indeed, if ϕ and ϕ are both isometric embeddings satisfying (2.1), then σ ∈ Ker(ψ − ψ ) ⊗ C, where, as above, ψ, ψ : T → T are the compositions of ϕ, ϕ with the projection to T .Hence, by assumption ψ = ψ .Using that ϕ and ϕ are both isometric embeddings allows one to conclude.
Remark 2.6.-The proof shows more.It allows one to control the number of isometric embeddings ϕ : T → T ⊕ Z • e satisfying (2.1).Indeed, up to a certain sign, such a ϕ is determined by a root of unity in O Kσ .Hence, the number of such maps ϕ is bounded by twice the number of roots of unity in O Kσ , which is bounded from above by a number depending only on [K σ : Q].
Remark 2.7.-The hypotheses can be relaxed a little.For example, one can consider isometric embeddings of T into a fixed finite index overlattice of T ⊕ Z • e (e.g. the saturation of T (S) ⊕ Z • e in H 2 (S, Z)).Indeed, instead of working with ϕ in the proof above one uses n • ϕ, where n is the index of the overlattice, and then replaces the left hand side of (2.2) by n 2 .Also, (e) 2 ∈ 2Z >0 can be replaced by (e) 2 ∈ K σ satisfying g((e) 2 ) ∈ R >0 for all embeddings g : K σ → C.
Consider the twistor space S → P 1 associated with a polarized K3 surface (S 0 , L 0 ).Proposition 2.8.-Assume that S 0 is a K3 surface with CM.Then there exist at most finitely many t ∈ P 1 such that the fibre S t is isomorphic to S 0 .In fact, there are at most finitely many t ∈ P 1 such that S t and S 0 are Fourier-Mukai partners.
The arguments below only use that the period field K σ is a CM field and, therefore, in particular algebraic.It is known, that under the additional assumption that S 0 is defined over Q the period field K σ is algebraic if and only if S 0 has CM, see [Tre15].
Proof.-Recall that a K3 surface with CM is a projective K3 surface S 0 for which the endomorphism field K = End Hdg (T (S 0 ) ⊗ Q) of endomorphisms of the rational Hodge structure T (S 0 ) ⊗ Q given by the transcendental lattice T (S 0 ) is a CM field with dim K (T (S 0 ) ⊗ Q) = 1, see [Huy16, Chapter 3] for references.It is known that any K3 surface with CM is defined over Q, but this will not be used in the argument.
Let K σ be the field generated by the periods σ i of S 0 as above.We claim that K σ ⊂ K.For this it is enough to show that (γ.σ) ∈ K for all γ ∈ T (S 0 ).As dim K (T (S 0 ) ⊗ Q) = 1 by assumption, T (S 0 ) is contained in the Q-span of all α(γ 1 ), α ∈ K ⊂ End(T (S 0 ) ⊗ Q), where γ 1 is as above the first vector of a basis γ i ∈ T (S 0 ) with (σ.γ 1 ) = 1.Hence, for γ ∈ T (S 0 ), (σ.γ) is a Q-linear combination of numbers of the form (σ.α(γ 1 )), α ∈ K.As (σ.α(γ 1 )) = (α σ.γ 1 ) = α ∈ K, this proves K σ ⊂ K. (Recall that α → α is an automorphism of the CM field K which under any embedding corresponds to complex conjugation.) Let us now turn to the proposition itself.Clearly, it is enough to show the finiteness of Fourier-Mukai partners in a twistor family.Suppose that a twistor fibre S t is derived equivalent to S 0 , i.e. that there exists an exact linear equivalence D b (S t ) D b (S 0 ) between their derived categories.Hence, in addition to the identification of lattices H 2 (S t , Z) = H 2 (S 0 , Z) induced by the twistor diffeomorphism S S 0 × P 1 , any chosen equivalence D b (S 0 ) ∼ → D b (S t ) provides us with an additional Hodge isometry between the transcendental lattices T (S 0 ) ∼ → T (S t ).The composition of the two induces an isometric embedding ϕ : , where e := c 1 (L 0 ).Hence, ϕ(T (S 0 )) is contained in the saturation of T (S 0 ) ⊕ Z • e ⊂ H 2 (S 0 , Z).For simplicity assume that ϕ : T (S 0 ) → T (S 0 ) ⊕ Z • e, but see Remark 2.7.
According to Lemma 2.5, under our assumptions there exist only finitely many such ϕ.Here we use that the transcendental lattice is general, for it is the minimal primitive sublattice containing σ in its complexification.As ϕ(σ) determines t up to sign, only finitely many fibres S t are derived equivalent to S 0 .
Remark 2.9.-Using Remark 2.6, we conclude that for a polarized K3 surface (S 0 , L 0 ) with CM one can bound the cardinality of the two finite sets Note however that infinitely many of the fibres S t may come with a polarization L t yielding infinitely many points (S t , L t ) in the moduli space of polarized K3 surfaces M d of fixed degree, which could be loosely phrased as saying that a twistor line usually intersects the quasi-projective moduli space M d in infinitely many points, only that the underlying K3 surfaces will not be isomorphic to each other.

2.3.
It may be instructive to look at K3 surfaces S 0 of maximal Picard number ρ(S 0 ) = 20.Those are known to have CM, cf.[Huy16,Remark 3.3.10].In this case, the arguments simplify.Indeed, the transcendental lattice T (S 0 ) is then positive definite (of rank two) and there exist only finitely many isometric embeddings of T (S 0 ) into any other fixed positive definite lattice, e.g.T (S 0 ) ⊕ Z • e.So finiteness in Lemma 2.5 follows directly.
K3 surfaces of maximal Picard rank can also be used to construct examples of twistor families with isomorphic distinct fibres.Indeed, if S 0 is a K3 surface with T (S 0 ) Z(d) ⊕2 with orthogonal basis e 1 , e 2 and such that there exists a polarization L of degree d for which T (S 0 ) ⊕ Z • L ⊂ H 2 (S 0 , Z) is saturated, then the two fibres S 1 , S 2 corresponding to T (S i ) = e ⊥ i have both transcendental lattices isomorphic to T (S 0 ).Therefore, by Orlov's result, D b (S i ) D b (S 0 ) and, as ρ(S i ) = 20, in fact S i S 0 .
A closer inspection of this case also reveals that Proposition 2.8 will be difficult to strengthen.For example, one could ask how many of the fibres S t are isogenous to S 0 , i.e. such that there exists a Hodge isometry H 2 (S, Q) H 2 (S t , Q) (or, equivalently, a Hodge isometry T (S) ⊗ Q T (S t ) ⊗ Q), or how many of them have the same Chow motive h(S 0 ) h(S t ), see [Huy17] for the relation between the two notions.However, finiteness fails in these settings.Indeed, for ρ(S 0 ) = 20 the projective fibres S t are up to conjugation uniquely determined by primitive classes e t in (the saturation of) T (S 0 ) ⊕ Z • e ⊂ H 2 (S 0 , Z).It is essentially (e t 1 ) 2 /(e t 2 ) 2 ∈ Q * /Q * 2 that decides whether S t 1 and S t 2 are isogenous.So one will usually have infinitely many fibres S t that are isogenous to S 0 and infinitely many that are not.

Finiteness via Kuga-Satake and abelian varieties
An approach to the finiteness of polarizations of fixed degree on an abelian variety A 0 modulo the action of Aut(A 0 ) similar to the one presented in Section 1 can be worked out.It provides a new proof of the classical result of Narasimhan and Nori [NN81], cf.[Mil86,Section 18].Note that the finiteness of A g,d (A 0 ) ⊂ A g,d can be quickly reduced to the case of principally polarized abelian varieties via Zarhin's trick [Zar85] which provides a (non-canonical) quasi-finite morphism A g,d → A 8g := A 8g,1 , (A, L) → ((A × Â) 4 , L) to the moduli space of principally polarized abelian varieties, see e.g. the account in [OS18, Section 4.1].
The question how many principal polarizations an abelian variety can admit has been studied by Lange [Lan87], who in particular describes bounds for the cardinality of A g (A 0 ) ⊂ A g for A 0 with real multiplication.Further results for products of elliptic curves can be found in [Hay68,Lan06], where, for example, it is shown that |A 2 (E 1 × E 2 )| is unbounded for isogenous elliptic curves E 1 , E 2 without complex multiplication.See also [How01,How05].
We shall now indicate an alternative proof of Theorem 0.1 based on the Kuga-Satake construction which reduces the problem to the finiteness for abelian varieties.For simplicity, we restrict to the case of K3 surfaces and leave the necessary modifications in the higher-dimensional case to the reader.
Starting with the Hodge structure of weight two H 2 (S, Z) L-pr of an arbitrary polarized K3 surface (S, L), the Kuga-Satake construction produces an abelian variety KS(S, L) of dimension g = 2 19 .By choosing a pair of positive orthogonal vectors in H 2 (S, Z) L-pr , e.g. e 1 + f 1 , e 2 + f 2 in the two copies of the hyperbolic plane U in the decomposition H 2 (S, Z) L-pr E 8 (−1) ⊕2 ⊕ U ⊕2 ⊕ Z(−d), one can define a polarization on KS(S, L) of a fixed degree d , cf. [Huy16, Chapter 4].Suppressing the choice of finite level structures, this leads to a morphism which is known to be quasi-finite, see [And96,Riz10] and [Mau14, Proposition 5.10].The Kuga-Satake construction applies to any Hodge structure of K3 type, in particular to the transcendental lattice T (S) of a K3 surface, which is independent of the polarization, and to T (S) ⊕ L ⊥ .Here, L ⊥ ⊂ NS(S) denotes the primitive sublattice orthogonal to an ample L ∈ NS(S).Moreover, KS(T (S) ⊕ L ⊥ ) KS(T (S)) 2 ρ(S)−1 .
Its degree depends only on the index of ι i .There may be infinitely many polarizations L i of fixed degree but only finitely many inequivalent ones under the action of O(NS(S)).Hence, the inclusions ι i have bounded index and, therefore, the KS(S, L i ) are all quotients of bounded degree of the abelian variety KS(T (S)) 2 ρ(S)−1 .This then shows the following result which allows one to deduce from the finiteness of A g,d (A 0 ) for abelian varieties [NN81,Mil86] the finiteness of M d (S 0 ) for K3 surfaces (and more generally of M d (X 0 ) for compact hyperkähler manifolds).
Corollary 3.1.-For any K3 surface S 0 the image of M d (S 0 ) ⊂ M d under the quasi-finite map (3.1) is contained in a finite union of sets A g,d (A i ) ⊂ A g,d , where the finitely many abelian varieties A i are quotients of the abelian variety KS(T (S)) 2 ρ(S)−1 of a fixed degree.
Using the finiteness of A g,d (A i ) [NN81] and the finiteness of the fibres of (3.1), this proves Theorem 0.1 once again.

Parametrization by special cycles and FM interpretation
The discussion in Section 1 can be understood more explicitly in the case of K3 surfaces, although most issues related to lattice theory remain, for example the decomposition of the moduli space according to the various ∈ N and the possible choices for the primitive embeddings N → Λ still occur (unless N is of small rank as in [Bea04]).

4.1.
Moduli spaces of lattice polarized K3 surfaces have first been studied in [Dol96] and later also in [Bea04].Dolgachev shows (using the global Torelli theorem) that the moduli space of 'ample N -polarized' K3 surfaces M N injects into the quotient Õ(T i )\ D T i .Here, Õ(T i ) ⊂ O(T i ) is the finite index subgroup of orthogonal transformations that extend to all of Λ by the identity on η i (N ).Note that in [Dol96] the embedding N → Λ is actually fixed so that only one period domain D i has to be considered and that ι(N ) is only requested to contain an ample class (but without fixing ∈ N or its image ι( )).So there exists a quasi-finite morphism M N,d → M N inducing injections M (N, ) → M N .In [Bea04] the rank of N is small enough to ensure that the embedding is actually unique, so that again only one period domain occurs.
For K3 surfaces there exists an interpretation of all points in the fibre of the map π i : M i N,d → O(T i ) \ D i through a very general (S 0 , ι 0 ).They correspond to Npolarized K3 surfaces whose transcendental part is Hodge isometric to T i T (S 0 ).Those are known to be the Fourier-Mukai partners of S 0 with Néron-Severi lattice isomorphic to NS(S 0 ), see [Huy16,Chapter 16] for references.
Hence, for K3 surfaces π i factorizes as where the degree of the second map is essentially the number of Fourier-Mukai partners for the very general K3 surfaces parametrized by M i N,d with fixed Néron-Severi lattice NS(S 0 ).

4.2.
As the sets M d (S 0 ) (or A g,d (A 0 ), M d (X 0 ), etc.) are finite, one may wonder whether they can be realized as fibres of a finite map from M d to some variety or space.Clearly, since |M d (S 0 )| for varying (S 0 , L 0 ) ∈ M d is unbounded (cf.Section 1.5), this cannot be true literally.However, we shall explain that this idea can be turned into a correct statement, which then also sheds light on the distributions of the M d (S 0 ).For this purpose it is more convenient to replace the sets M d (S 0 ) ⊂ M d by the sets Recall that for ρ(S 0 ) 12 the existence of a Hodge isometry (4.2) is in fact equivalent to the existence of an isomorphism S S 0 .In general, as M d (S 0 ) also the set M d (D b (S 0 )) is finite and the arguments in Section 1 in fact reprove this result.See [Huy16,Chapter 16] for references and further details and [HP13] proving finiteness of Fourier-Mukai orbits in the moduli space of quasi-polarized K3 surfaces.
be bounded.Then use the interpretation of Cl(K i ) as the set of Sl(2, Z)-equivalence classes of binary quadratic forms aX 2 + bXY + cY 2 of discriminant −p i = b 2 − 4ac or, equivalently, as the set of isomorphism classes of oriented positive definite lattices Γ of rank two with intersection matrix ( 2a b b 2c For a concrete example, consider an automorphism f : S ∼ → S of infinite order of a K3 surface S. Then consider the family π : X := Bl ∆ (S × S) σ → S × S p 1 → S, which over a point s ∈ S is the blow-up of S in s.Fix a sufficiently ample line bundle L on S and the induced π-ample line bundle p * 1 L(−E), M d .Let us spell this out for K3 surfaces.Define Λ d := ⊥ ⊂ Λ as the orthogonal complement of a primitive class in the K3 lattice Λ with ( ) 2 = d.Then M d is an open subset of the arithmetic quotient Õ(Λ d )\D of the period domain D ⊂ P(Λ d ⊗C) (viewed as the set of positive oriented planes in Λ d ⊗ R) by the stabilizer of , cf. [Huy16, Chapter 6].Now, for any α ∈ Λ d with (α) 2 > 0 consider the positive cone C α ⊥ ⊂ α ⊥ ⊗ R. Note that α ⊥ is of signature (1, 19).The image L α ⊂ M d of the natural map C α ⊥ /R * → D → Õ(Λ d ) \ D, that sends β ∈ C α ⊥ to the plane spanned by α and β, or rather its intersection with the open set M d , describes the set of all polarized K3 surfaces (S, L) with α ∈ (H 2,0 ⊕H 0,2 )(S, Z).Each of the countably many L α ⊂ M d is of real dimension 19 and isotropic with respect to the natural symplectic form on M d .Compare this to [BL16, Remark 5.7].
(2.3) |{t ∈ P 1 | S t S 0 }| and |{t ∈ P 1 | D b (S t ) D b (S 0 )}| by a constant c(K) only depending on the CM field K = End Hdg (T (S 0 ) ⊗ Q).In fact, as [K : Q] = rk T (S 0 ) 21, the Euler function ϕ(m) of the m-th roots of unity that can occur in the proof of Lemma 2.5 is bounded by 21 and hence m 66. Taking complex conjugation into account, this shows that the numbers in (2.3) are universally bounded by 132.

M
d (D b (S 0 )) ⊂ M d of all (S, L) ∈ M d for which there exists an exact linear equivalence (4.1) D b (S 0 ) D b (S) between the bounded derived categories of coherent sheaves on S and S 0 .According to a result of Mukai and Orlov, the condition is equivalent to the existence of a Hodge isometry (4.2) T (S) T (S 0 ).