Limit mixed Hodge structures of hyperkähler manifolds
aa r X i v : . [ m a t h . AG ] J un LIMIT MIXED HODGE STRUCTURES OF HYPERKÄHLER MANIFOLDS
ANDREY SOLDATENKOV
Abstract.
This note is inspired by the work of Deligne [De]. We study limit mixed Hodge structuresof degenerating families of compact hyperkähler manifolds. We show that when the monodromy actionon H has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups areof Hodge-Tate type. Introduction
It is well-known that for the study of mirror symmetry it is important to consider families of Calabi-Yau varieties with “maximal degeneration” at the special fibre. There are several slightly different waysto give the definition of maximal degeneration (cf. [Mor, Definition 3], [KS, Definition 1], [De]), notall of them being equivalent to each other. In any of these definitions the condition is Hodge-theoretic,and concerns the limiting behavior of the corresponding variation of Hodge structures. Presumably thestrongest condition was suggested by Deligne [De]: a degeneration of Hodge structures is called maximal,if the corresponding limit mixed Hodge structure is of Hodge-Tate type, i.e. it is an iterated extension ofdirect sums of Z ( k ), k ∈ Z .We study projective degenerations of compact simply-connected hyperkähler manifolds over the unitdisc (see Definition 2.1). The main result (Theorem 3.8) states the following: if the monodromy operator γ acting on H is unipotent of maximal index, i.e. ( γ − id) = 0 and ( γ − id) = 0, then the limit mixedHodge structures on H k are of Hodge-Tate type for all k . We deduce this result from the generalizedKuga-Satake construction [KSV], [SS]. The condition of maximal unipotency of monodromy is a prioryweaker than maximality in the sense of Deligne. Our result shows that these two notions coincide in thecase of hyperkähler manifolds.The key step for understanding the limit mixed Hodge structures of degenerations is the description ofthe monodromy action on the cohomology ring. Using the results of Verbitsky [Ve3, Theorem 3.5(iii)], weshow that monodromy action on the full cohomology ring is essentially determined by its action on H (see Proposition 3.5).In the case of maximal degenerations of compact hyperkähler manifolds, one can determine the unipo-tency indices of the monodromy action on H k . For even k this was done in [KLSV, Proposition 6.18], seealso the paper of Nagai [Nag] for the general discussion and related results. We compute the unipotencyindices for odd k , see Proposition 3.15. This result applies to degenerations of generalized Kummer typemanifolds, since they have non-trivial cohomology groups in odd degrees.In section 4 we discuss existence of maximal degenerations, showing that such degenerations exist inevery deformation equivalence class of compact hyperkähler manifolds with b > Date : June 13, 2019.2010
Mathematics Subject Classification. primary 14D06, 14D07; secondary 14D05.The author was supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG(German Research Foundation). esult has already appeared in the preprint [To]. We provide a simple independent argument, showingthat one can always find a nilpotent orbit (see Definition 4.3) with maximally unipotent monodromy thatis induced by a projective degeneration of hyperkähler manifolds.2. Degenerations with maximally unipotent monodromy
In this section we recall some well-known facts about compact hyperkähler manifolds and their perioddomains, for an overview see [Hu1]. We also recall necessary facts about degenerations and limit mixedHodge structures.2.1.
Hyperkähler manifolds.
Recall that a compact Kähler manifold X is called simple hyperkähler,or irreducible holomorphic symplectic (IHS), if it is simply-connected and H ( X, Ω X ) is spanned by asymplectic form. In what follows we will always assume that X is simple hyperkähler of complex dimension2 n .Let V Z = H ( X, Z ) and V = V Z ⊗ Q . Note that V Z is torsion-free, because X is simply-connected.Recall that there exists a non-degenerate form q ∈ S V ∗ and a constant c X ∈ Q , such that for all h ∈ H ( X, Q ) ≃ V we have q ( h ) n = c X h n , where we use the cup product in cohomology. The form q iscalled Beauville-Bogomolov-Fujiki (BBF) form. We normalize q to make it integral and primitive on V Z ,and such that q ( h ) > h . Then q has signature (3 , b ( X ) − D ⊂ P ( V C ) be the quadric defined by q , and D = { x ∈ ˆ D | q ( x, ¯ x ) > } . Given an element h ∈ V Z with q ( h ) > V h = { v ∈ V | q ( h, v ) = 0 } , ˆ D h = ˆ D ∩ P ( V h C ), D h = D ∩ P ( V h C ). Then ˆ D h is the extended period domain and D h is the period domain for polarized Q -Hodge structures of K3 typeon ( V h , q ).The group G R = O( V h R , q ) acts transitively on D h . After fixing a base point in D , we get an isomorphism D h ≃ G R /K , where K is a compact subgroup. Analogously, G C = O( V h C , q ) acts transitively on ˆ D h . Thediscrete group O( V h Z , q ) ⊂ G R acts on D h properly discontinuously, and according to Baily-Borel thequotient D h / O( V h Z , q ) is a quasi-projective variety. We can pass to a finite index torsion-free subgroupΓ ⊂ O( V h Z , q ), so that the quotient D h / Γ is moreover smooth.2.2.
Degenerations.
Denote: ∆ = { z ∈ C | | z | < } , ∆ ∗ = ∆ \{ } . Given a morphism π : X → ∆ and t ∈ ∆ we write X t = π − ( t ). Definition 2.1.
A degeneration of X is a flat proper morphism of complex-analytic spaces π : X → ∆ ,such that: π is smooth over ∆ ∗ ; the fibre X t is deformation equivalent to X for all t ∈ ∆ ∗ ; the monodromyaction on the second cohomology of X t is unipotent and non-trivial. The degeneration is called projective,if π is a projective morphism.Remark . The condition of unipotency is almost automatic: it follows from a theorem of Borel (see[Sch, Lemma 4.5]), that monodromy of any family becomes unipotent after we pass to a finite ramifiedcover of ∆. Non-triviality of monodromy excludes the case when π is smooth over the whole ∆. Note thatwe do not require any of the smooth fibres of π to be isomorphic to X , but only deformation-equivalentto it. One may think that our degenerations represent “boundary points” of the connected component ofthe moduli space that contains X .Let π : X → ∆ be a projective degeneration of X . Denote by π ′ the restriction of π to π − (∆ ∗ ) andconsider the local system V = R π ′∗ Z over ∆ ∗ . Fix a base point t ∈ ∆ ∗ , identify H ( X t , Z ) with V Z andlet h ∈ V Z be the class of the polarization. Then V is a variation of Hodge structures (VHS) with fibre Z , h determines a sub-VHS in it, and q defines a bilinear pairing on V . Let V h be the q -orthogonalcomplement of h ; it is a VHS with fibre V h Z polarized by q .Let ˜∆ = { z ∈ C | Im( z ) > } and τ : ˜∆ → ∆ ∗ , z e πiz be the universal covering. The pull-back τ ∗ V h Z is a trivial local system and the VHS on it defines a period map ˜ ϕ : ˜∆ → D h . By our definition ofdegeneration, the monodromy transformation γ ∈ Aut( V h , q ) ≃ O( V h Z , q ) is of the form γ = e N , where N ∈ so ( V h , q ) is nilpotent of index 2 or 3. This restriction on the index of nilpotency follows from thegeneral statement [Sch, Theorem 6.1]. Definition 2.3.
Degenerations of X with N of index will be called maximally unipotent.Remark . There exists a different terminology, used mainly in the case of degenerations of K3 surfaces:the degeneration is of “type II” and “type III” when N has nilpotency index 2, respectively 3, see e.g.[Ku].We recall the results of Schmid [Sch] about limit mixed Hodge structures (MHS). The period map ˜ ϕ satisfies the relation ˜ ϕ ( z +1) = γ ˜ ϕ ( z ). Define the map ˜ ψ : ˜∆ → ˆ D h , z e − zN ˜ ϕ ( z ). Then ˜ ψ ( z +1) = ˜ ψ ( z ),and ˜ ψ descends to a map ψ : ∆ ∗ → ˆ D h . According to the nilpotent orbit theorem [Sch, Theorem 4.9], ψ extends over the puncture, and the point ψ (0) determines a decreasing filtration F • lim on V h C . Anotherfiltration W • , increasing and defined over Q , is induced on V h by the nilpotent operator N . It followsfrom the SL -orbit theorem that these two filtrations and the form q determine a polarized mixed Hodgestructure on V h , see [Sch, Theorem 6.16]. Definition 2.5.
A mixed Hodge structure ( U, W • , F • ) is of Hodge-Tate type, if its Hodge numbers satisfythe condition h p,q = 0 for p = q . Equivalently, gr W p +1 U = 0 , and gr W p U is a pure Hodge structures of type ( p, p ) for all p . It is easy to check (see e.g. [Ku]) that for a maximally unipotent degeneration of X the limit MHS( V h , W • , F • lim ) on the second cohomology is of Hodge-Tate type.This finishes the discussion of the limit MHS on the second cohomology of X . Next, one can apply theabove constructions to higher degree cohomology groups. To study their behavior under degeneration,we use the relation between Hodge structures on higher cohomology groups and on H . This will beexplained in the next section.3. Limit mixed Hodge structures of maximally unipotent degenerations
In this section we fix X , V Z , V and q as above. We consider a projective degeneration π : X → ∆ of X , and assume without loss of generality that X ≃ X t for a fixed base point t ∈ ∆ ∗ . We let h ∈ V Z bethe class of the polarization.3.1. The Mukai extension and the mapping class group.
Consider the graded Q -vector space˜ V = h e i ⊕ V ⊕ h e i , where e i is of degree i , and V is in degree 2. We introduce on ˜ V a quadratic form ˜ q that is determined by the following conditions: ˜ q | V = q , e and e are isotropic and orthogonal to V andspan a hyperbolic plane, so that ˜ q ( e , e ) = 1. We call ( ˜ V , ˜ q ) the Mukai extension of ( V, q ).Consider the graded Lie algebra so ( ˜ V , ˜ q ) and denote by Ξ the generator of the orthogonal algebra of h e , e i , such that Ξ e = e , Ξ e = − e . Denote by W the Weil operator that induces the Hodge structureon V , i.e. it acts on V p,q as multiplication by i ( p − q ). It is clear that Ξ , W ∈ so ( ˜ V , ˜ q ). e recall that there exists a representation of graded Lie algebras so ( ˜ V , ˜ q ) → End( H • ( X, Q )), suchthat: the action of Ξ induces the cohomological grading on H • ( X, Q ); the action of W induces the Hodgestructures on H k ( X, Q ) for all k . For the proof we refer to [Ve1], [LL] or [KSV, Theorem A.10].Recall also, that so ( V, q ) acts on H • ( X, Q ) by derivations, see [Ve1, Corollary 13.5]. It acts triviallyon all Pontryagin classes of X , since the Pontryagin classes stay of Hodge type ( p, p ) on all deformationsof X . Denote by Aut P ( X ) ⊂ GL( H • ( X, Q )) the group of algebra automorphisms that fix the Pontryaginclasses. We obtain a homomorphism of algebraic groups α : Spin( V, q ) → Aut P ( X ). Let us denote byAut + ( X ) the image of α .It was shown in [HS] (see also [Hu2]), that R X p td( X ) >
0. Here td( X ) denotes the total Toddclass of X . Since all odd Chern classes of X vanish, td( X ) can be expressed as a universal polynomialin the Pontryagin classes. It follows that all elements of Aut P ( X ) act trivially on H n ( X, Q ), where2 n = dim C ( X ).Consider the action of Aut P ( X ) on H ( X, Q ). Note that the form q is uniquely up to a sign determinedby the multiplicative structure of the cohomology ring. It follows from the above discussion that the actionof Aut P ( X ) preserves q (see [Ve3, Theorem 3.5(i)]). Hence we have a homomorphism β : Aut P ( X ) → O( V, q ).We get the following commutative diagram of algebraic groups, where the maps α ′ and β ′ are isogenies:(3.1) Spin( V, q ) Aut + ( X ) Aut P ( X )SO( V, q ) O(
V, q ) α ′ β ′ β Given a Q -algebraic group G we will denote by G Q the group of its rational points. Lemma 3.1.
Let Γ A be an arithmetic subgroup of Aut P ( X ) Q . Then Γ + A = Γ A ∩ Aut + ( X ) Q is of finiteindex in Γ A .Proof. We first outline the idea of the proof. We do not know a priory that the subgroup Aut + ( X ) is offinite index in Aut P ( X ), so we can not prove the statement directly. We will instead consider the imagesof Γ + A and Γ A under the maps β ′ and β . These images are arithmetic subgroups of O( V, q ) Q , hence theyare commensurable, which is enough to deduce the claim of the lemma.The group Γ + A is an arithmetic subgroup of Aut + ( X ) Q , because Γ A is an arithmetic subgroup ofAut P ( X ) Q . Let us denote by Γ and Γ +2 the images of Γ A in O( V, q ) Q and of Γ + A in SO( V, q ) Q respectively.By [Bor, Theorem 8.9] Γ +2 is an arithmetic subgroup of SO( V, q ) Q . It is also an arithmetic subgroup ofO( V, q ) Q , since SO( V, q ) Q is an index two subgroup in it. By [Bor, Corollary 7.13] Γ is contained in anarithmetic subgroup of O( V, q ) Q . Since Γ also contains an arithmetic subgroup Γ +2 , it is arithmetic itself.We will use the following observation. Let φ : G → G be a surjective homomorphism of groups withfinite kernel, and let H ⊂ G be a subgroup. If φ ( H ) is of finite index in G , then H is of finite index in G . Let us apply the observation to G = Γ A , G = Γ and H = Γ + A . The proof of [Ve3, Theorem 3.5(iii)]shows that the kernel of β : Γ A → Γ is finite (note that the cited proof applies to arbitrary arithmeticsubgroups of Aut P ( X ) Q ). The subgroup β (Γ + A ) = Γ +2 has finite index in Γ , because both are arithmeticsubgroups of O( V, q ) Q . We conclude that Γ + A is of finite index in Γ A . (cid:3) Remark . The lemma can be applied to the arithmetic subgroup Γ A = Aut P ( X ) Z of automorphismsthat preserve the integral cohomology classes. Let MC( X ) = Diff( X ) / Diff ( X ) be the mapping class group f X . Here Diff( X ) is the group of diffeomorphisms of X , and Diff ( X ) is the subgroup of diffeomorphismsisotopic to the identity. We have a natural homomorphism MC( X ) → Aut P ( X ) Z . The above lemma showsthat the action of MC( X ) on the cohomology algebra can essentially be recovered from its action on H ,up to some elements of finite order, bounded by the index of Aut P ( X ) + Z in Aut P ( X ) Z .3.2. The Kuga-Satake construction.
We recall the main result of [KSV]. To a Hodge structure V ofK3 type one can associate the Kuga-Satake Hodge structure of abelian type. It is constructed as follows.Let H = C l ( V, q ) be the Clifford algebra and let v ∈ V C be the generator of V , . Define H , − to bethe right ideal vH C (see [SS, Lemma 3.3]), and let H − , = H , − . One can check that this defines aHodge structure on H . Let H h be analogously defined Hodge structure for V h . Then H ≃ ( H h ) ⊕ , andone can check that H is polarized, although the polarization is not canonical. More precisely, fix a pairof elements a , a ∈ V h , such that q ( a ) > q ( a ) > q ( a , a ) = 0. Let a = a a ∈ C l ( V h , q ) and ω ( x, y ) = Tr( xa ¯ y ), where x, y ∈ C l ( V h , q ), the map y ¯ y is the canonical anti-involution, and Tr is thetrace on the Clifford algebra (see e.g. [KSV, Proposition 4.2]). Then either ω or − ω defines a polarizationof H h , moreover ω is Spin( V h , q )-invariant. When we apply the Kuga-Satake construction to a VHS ofK3 type, the monodromy operator lies in Spin( V h , q ) (see [SS, Section 3.1]), hence the form ω is alwaysmonodromy-invariant.Note that H is canonically an so ( V, q )-module, and the Hodge structure on it is induced by the actionof the Weil operator W . The following theorem was proved in [KSV, Theorem 4.1] Theorem 3.3.
There exists a structure of graded so ( ˜ V , ˜ q ) -module on Λ • H ∗ that extends the canonical so ( V, q ) -module structure. Moreover, there exists an integer m > and an embedding of so ( ˜ V , ˜ q ) -modules (3.2) H • +2 n ( X, Q ) ֒ → Λ • +2 d ( H ∗⊕ m ) , where n = dim C ( X ) and d = m dim Q ( H ) . In particular, for i = − n, . . . , n we get an embedding ofHodge structures H i +2 n ( X, Q ( n )) ֒ → Λ i +2 d ( H ∗⊕ m )( d ) . Remark . The shifts in the cohomological grading in the above statement are necessary to make thegrading compatible with the action of the element Ξ ∈ so ( ˜ V , ˜ q ). The statement about the embedding ofHodge structures includes the appropriate Tate twists.3.3. Main result.
We go back to the degeneration π : X → ∆. The monodromy acting on H ( X, Q ) is γ = e N , where N ∈ so ( V, q ) ⊂ so ( ˜ V , ˜ q ). Let us denote by δ ∈ GL( H • ( X, Q )) the monodromy operator forthe full cohomology algebra. Proposition 3.5.
There exists an integer k > , such that δ k = e kN , where e kN acts on H • ( X, Q ) viathe representation Spin(
V, q ) → Aut P ( X ) .Proof. The monodromy operator δ is induced by a diffeomorphism of X t for a base point t ∈ ∆ ∗ . Henceit is contained in the arithmetic subgroup Γ A = Aut P ( X ) Z of Aut P ( X ) Q . The claim follows from Lemma3.1 applied to this subgroup. (cid:3) We get the following immediate consequence, that recovers Corollary 3.2 from [KLSV]:
Corollary 3.6.
If the monodromy action on H ( X, Q ) is trivial, then its action on H • ( X, Q ) is of finiteorder. Next we compare the limit MHS on X and the Kuga-Satake abelian variety. roposition 3.7. There exists an integer m > and an embedding of mixed Hodge structures ( H • +2 n ( X, Q ( n )) , ˜ W • , ˜ F • lim ) ֒ → (Λ • +2 d ( H ∗⊕ m )( d ) , W • , F • lim ) , where n = dim C ( X ) and d = m dim Q ( H ) .Proof. We use the same convention with the shift of cohomological grading as in Theorem 3.3, see theremark after that theorem. In particular, the Hodge filtration on H • +2 n ( X, Q ( n )) has non-trivial gradedcomponents in degrees − n, . . . , n .The limit mixed Hodge structures do not change if we replace the monodromy operator by its power.Thus we may use Proposition 3.5 and assume that δ = e N , where the exponential is viewed as an elementof Spin( V, q ). This implies that the embedding from Theorem 3.3 is compatible with the weight filtrations,since they are both induced by the action of N ∈ so ( V, q ).Next we deal with the limit Hodge filtrations. Let us denote by D X and ˆ D X the period domain,respectively the extended period domain for the h -polarized Hodge structures on H • ( X, Q ). Analogously, D KS and ˆ D KS will denote the period domain, respectively the extended period domain for the Hodgestructures on Λ • ( H ∗⊕ m ) polarized by a fixed form ω as above. Both ˆ D X and ˆ D KS are closed subvarietiesof certain flag varieties, and D X , D KS are their open subsets (see [Sch] for the description of perioddomains as subvarieties of flag varieties).The variety ˆ D X carries a universal family of holomorphic bundles that determine the Hodge filtration: H • ( X, Q ) ⊗ O ˆ D X = ˜ F − n ⊃ ˜ F − n +1 ⊃ . . . ⊃ ˜ F n ⊃ ˜ F n +1 = 0 . Analogously, over ˆ D KS we have a family of subbundlesΛ • ( H ∗⊕ m ) ⊗ O ˆ D KS = F − d ⊃ F − d +1 ⊃ . . . ⊃ F d ⊃ F d +1 = 0 . Let p X and p KS denote the two projections from ˆ D X × ˆ D KS to the factors. For every i = − n, . . . , n consider the morphism of vector bundles η i : p ∗ X ˜ F i → p ∗ KS ( F − d / F i )obtained as the composition of three morphisms: the embedding p ∗ X ˜ F i ֒ → H • ( X, Q ) ⊗ O ˆ D X × ˆ D KS , theembedding from Theorem 3.3 and the projection to the quotient Λ • ( H ∗⊕ m ) ⊗O ˆ D X × ˆ D KS → p ∗ KS ( F − d / F i ).Denote by Z the closed subscheme of ˆ D X × ˆ D KS where all η i vanish. The points of Z correspond to suchpairs of filtrations that the embedding from Theorem 3.3 is compatible with them.After passing to the universal cover of the punctured disc, we get two period maps ˜ ϕ X : ˜∆ → D X and˜ ϕ KS : ˜∆ → D KS . Since the Hodge structures on H • ( X, Q ) and Λ • ( H ∗⊕ m ) are both determined by theaction of the Weil operators W ( z ) ∈ so ( V, q ), z ∈ ˜∆, the embedding from Theorem 3.3 is a morphism ofHodge structures. This means that the product of ˜ ϕ X and ˜ ϕ KS gives a map ˜ ϕ : ˜∆ → Z ⊂ ˆ D X × ˆ D KS .Consider now the twisted period maps ˜ ψ X : ˜∆ → ˆ D X and ˜ ψ KS : ˜∆ → ˆ D KS , where ˜ ψ X ( z ) = e − zN ˜ ϕ X ( z )and ˜ ψ KS ( z ) = e − zN ˜ ϕ KS ( z ). Let ˜ ψ : ˜∆ → ˆ D X × ˆ D KS be their product. Since the subscheme Z isSpin( V, q )-invariant by construction, we have ˜ ψ : ˜∆ → Z .By a theorem of Schmid [Sch, Theorem 4.9] there exists a limit lim Im z → + ∞ ˜ ψ ( z ). The corresponding pointof ˆ D X × ˆ D KS determines the pair of limit Hodge filtrations ˜ F • lim and F • lim . Since the subscheme Z isclosed, the limit lies in Z . We conclude that the limit Hodge filtrations are compatible. (cid:3) Theorem 3.8.
Let X be a simple hyperkähler manifold and let π : X → ∆ be a maximally unipotentprojective degeneration of X . Then the limit mixed Hodge structures on H k ( X, Q ) are of Hodge-Tate typefor all k . roof. Mixed Hodge structures of Hodge-Tate type form a tensor subcategory inside the abelian categoryof MHS. Hence by Proposition 3.7 it suffices to check that the limit MHS of the variation of Kuga-SatakeHodge structures (
H, W • , F • lim ) is of Hodge-Tate type. This follows from [SS, proof of Theorem 1.2(3)]. (cid:3) The knowledge of the limit MHS provides some information about cohomology of the central fibre, atleast if the degeneration is semistable.Let Y be a quasi-projective variety. Recall, that cohomology groups of Y carry functorial mixed Hodgestructures. If Y is projective, the weights of the MHS on H k ( Y, Q ) lie in the range 0 , . . . , k for k dim C ( Y )and 2 k − C ( Y ) , . . . , k for k > dim C ( Y ). Definition 3.9.
We will say that the mixed Hodge structure on H k ( Y, Q ) is semi-pure, if the inducedmixed Hodge structure on W k − H k ( Y, Q ) is of Hodge-Tate type. Corollary 3.10.
In the setting of Theorem 3.8, assume that X is a reduced divisor with simple normalcrossings. Then the mixed Hodge structures on H k ( X , Q ) are semi-pure for all k .Proof. Consider the Clemens-Schmid exact sequence of MHS, where ν = (2 πi ) − N : . . . −→ H k X ( X , Q ) −→ H k ( X , Q ) −→ H k lim ( X, Q ) ν −→ H k lim ( X, Q )( − −→ . . . It follows from Poincaré duality that the MHS on H k X ( X , Q ) has weights > k . Hence the MHS on W k − H k ( X , Q ) is determined by the limit MHS of the degeneration. The claim now follows from Theorem3.8. (cid:3) Unipotency indices of the monodromy action on higher cohomology groups.
It was ob-served in [KLSV, Proposition 6.18], that for maximally unipotent degenerations of hyperkähler manifoldsthe index of unipotency of the monodromy action on H k ( X, Q ) equals 2 k + 1, where k = 1 , . . . , n andas before 2 n = dim C ( X ). We will explain below, that it is also possible to determine the index of unipo-tency for odd degree cohomology groups, see Proposition 3.15. This applies, in particular, to maximaldegenerations of generalized Kummer type manifolds.In this subsection we will briefly write H • for H • ( X, C ) considered as an so ( ˜ V C , ˜ q )-module (see section3.1). We will use the highest weight theory for the orthogonal Lie algebra (see e.g. [Bou, Chapter VIII,§13]). Let us fix two elements ξ , ξ ∈ so ( ˜ V C , ˜ q ) that define the Hodge bigrading on H • . More precisely, ξ acts on H r,s as multiplication by ( r + s ) − n , and ξ as multiplication by ( s − r ). Next we choosea Cartan subalgebra ˜ h ⊂ so ( ˜ V C , ˜ q ) that contains these two elements and fix a basis ˜ h = h ξ , ξ , . . . , ξ l i ,where l = ⌊ dim V ⌋ . Note that h = ˜ h ∩ so ( V C , q ) = h ξ , . . . , ξ l i is a Cartan subalgebra of so ( V C , q ). Let ε i denote the dual basis: ˜ h ∗ = h ε , . . . , ε l i .We recall from loc. cit. the expressions for positive roots and fundamental weights. In the case of odddim V , the set of positive roots in ˜ h ∗ is R + = { ε i | i l } ∪ { ε i ± ε j | i < j l } ; the fundamentalweights are: ̟ i = ε + . . . + ε i , i = 0 , . . . , l − ̟ l = ( ε + . . . + ε l ). The representation with highestweight ̟ l is the spinor representation.In the case when dim V is even, we always have l >
2, since b ( X ) >
3. Then R + = { ε i ± ε j | i The so ( V C , q ) -module H is the direct sum of several copies of spinor or semi-spinorrepresentations.Proof. We assume that H = 0. Since H , = H , = 0, the vector space H is the direct sum of twoeigenspaces of ξ with eigenvalues ± . Assume that H contains an irreducible subrepresentation withhighest weight a ̟ + . . . + a l ̟ l , a i ∈ Z > . If dim V is odd, ξ acts on the highest weight vector as thescalar a + . . . + a l − + a l . This is only possible when a = . . . = a l − = 0 and a l = 1. If dim V is even, ξ acts as a + . . . + a l − + ( a l − + a l ). This is only possible when a = . . . = a l − = 0 and either a l − = 1, a l = 0, or a l − = 0, a l = 1. In both cases we have either spinor or semi-spinor representation. (cid:3) Lemma 3.12. Let W • ⊂ H • be an irreducible so ( ˜ V C , ˜ q ) -submodule. Then W ≃ W n − is an irreducible so ( V C , q ) -module.Proof. We can assume that W = 0. Then W k = 0 for all k , since otherwise W • would be a non-trivialsubrepresentation of W • . We also have W = H = 0, and it follows that the minimal eigenvalue of ξ is − n , with W being the corresponding eigenspace. We can find a vector w ∈ W that is oflowest weight with respect to so ( V C , q ) and h . Then w is also of lowest weight for so ( ˜ V C , ˜ q ) and ˜ h . Byour assumption, w is unique up to multiplication by a scalar. This implies irreducibility of W . Theisomorphism W ≃ W n − follows from Poincaré duality. (cid:3) Lemma 3.13. Let W • ⊂ H • be an irreducible so ( ˜ V C , ˜ q ) -submodule, such that W = 0 . Then the highestweight µ of W • is one of the following. If dim( V ) is odd, then µ = ( n − ̟ + ̟ l . If dim( V ) is even,then either µ = ( n − ̟ + ̟ l − or µ = ( n − ̟ + ̟ l .Proof. It follows from Lemma 3.11 and Lemma 3.12 that W n − is either spinor or semi-spinor represen-tation of so ( V C , q ). Thus µ is either of the form k̟ + ̟ l or k̟ + ̟ l − (when dim( V ) is even), for some k . Then ξ acts on the highest weight vector as k + , and since the highest weight vector is contained in W n − , we have k = n − (cid:3) Lemma 3.14. Assume that H ( X, C ) = 0 . Then H k +1 ( X, C ) for k = 1 , . . . , n − contains an so ( V C , q ) -submodule of highest weight ν , which can be one of the following. If dim( V ) is odd, then ν = ( k − ̟ + ̟ l ;if dim( V ) is even, then either ν = ( k − ̟ + ̟ l or ν = ( k − ̟ + ̟ l − .Proof. Let us assume that dim( V ) is odd, the other case being analogous. We pick an irreducible so ( ˜ V C , ˜ q )-submodule W • ⊂ H • with W = 0. We know from Lemma 3.13 that the highest weight of W • is µ = ( n − ̟ + ̟ l = n − ε + ( ε + . . . + ε l ). The set of weights of a representation is invariant withrespect to the Weyl group action. Since the transposition of ε and ε belongs to the Weyl group, theweight µ ′ = ε + n − ε + ( ε + . . . + ε l ) also belongs to W • . Let α = ε − ε be one of the positiveroots. Then µ ′ = µ − ( n − α . Consider the action of the sl -subalgebra corresponding to the root α .It follows from the representation theory of sl , that all the weights of the form µ − iα , i = 0 , . . . , n − elong to W • . The corresponding weight subspaces are contained in H n − − i . By restricting to so ( V C , q )we find that H i ≃ H n − − i contains a subrepresentation with highest weight i +12 ε + ( ε + . . . + ε l ).Setting k = i + 1 we get the result. (cid:3) Proposition 3.15. Assume that H ( X, Q ) = 0 . Consider a maximal degeneration of X , and let N denote the logarithm of the monodromy acting on H k +1 ( X, Q ) , where k = 1 , . . . , n − . Then N k − = 0 , N k = 0 .Proof. The fact that N k = 0 follows from the general result of Schmid [Sch, Theorem 6.1] and thevanishing of Hodge numbers h k +1 , ( X ) = h , k +1 ( X ) = 0.Let N denote the logarithm of the monodromy acting on H ( X, Q ). According to Proposition 3.5,we may assume that N is the image of N under the homomorphism so ( V, Q ) → End( H k +1 ( X, Q )) (seesection 3.1). Let us assume that dim( V ) is odd. By Lemma 3.14, it is enough to consider the representationof highest weight ( k − ̟ + ̟ l , and to prove that N k − acts non-trivially on it.We can choose two isotropic subspaces U = h e , . . . , e l i ⊂ V C and U ′ = h e ′ , . . . , e ′ l i ⊂ V C and anelement e l +1 orthogonal to them, so that q ( e i , e ′ j ) = 0 for 1 i < j l , q ( e i , e ′ i ) = 1, q ( e l +1 , e l +1 ) = 1and V C = U ⊕ U ′ ⊕ h e l +1 i . We may moreover assume (see [SS, proof of Proposition 4.1]) that thisdecomposition is compatible with N in the sense that N = e ′ ∧ ( e + e ′ ), where we use the identification so ( V C , q ) ≃ Λ V C . We also choose the Cartan subalgebra of so ( V C , q ) corresponding to this decomposition(see [Bou, Chapter VIII, §13]).Denote by P i the so ( V C , q )-module of highest weight i̟ . Then P i is a subrepresentation of S i V C , andit is generated by the highest weight vector e i . Note that N e = − e − e ′ , N e = − e ′ and by Leibnitz’srule N i ( e i ) = 0.Let S be the spinor representation. It can be described as Λ • U , on which e i act by exterior multiplicationand e ′ i act by contraction, i = 1 , . . . , l . The highest weight vector is u = e ∧ . . . ∧ e l ∈ Λ • U , and we seethat N u = e ∧ . . . ∧ e l = 0.The element e k − ⊗ u ∈ P k − ⊗ S has weight ( k − ̟ + ̟ l , hence it generates the representationwe are interested in. Leibnitz’s rule again implies that N k − ( e k − ⊗ u ) = 0. This proves the claim fordim( V ) odd. The case of even dimension is analogous. (cid:3) Existence of degenerations with maximal unipotent monodromy In this section we fix a hyperkähler manifold X as in section 2.1, and assume moreover that b ( X ) > X admits a projective degeneration in the sense of Definition 2.1, such that the monodromy operator γ ∈ O ( V Z , q ) is of the form γ = e N , N ∈ so ( V, q ) with N = 0, N = 0.The construction consists of two steps. First, we find a nilpotent operator N that satisfies the aboveconditions and prove that there exist sufficiently many nilpotent orbits (see Definition 4.3). Second, weshow that one can find a nilpotent orbit that corresponds to a projective degeneration of X .4.1. Nilpotent orbit. We fix V Z , V , q as in section 2.1. Recall that the signature of q is (3 , dim( V ) − h ∈ V , V h denotes its orthogonal complement. Lemma 4.1. Assume that dim( V ) > . Then there exist an element h ∈ V Z with q ( h ) > and anendomorphism N ∈ so ( V h , q ) , such that N = 0 , N = 0 and the restriction of q to Im( N ) is semi-positive with one-dimensional kernel. roof. By Meyer’s theorem there exists a vector v ∈ V , such that q ( v ) = 0. An elementary argumentshows that v is contained in a hyperbolic plane: v = 1 / v + v ) for some v , v ∈ V with q ( v ) = 1, q ( v ) = − q ( v , v ) = 0. The restriction of q to the subspace V ′ = h v , v i ⊥ has signature (2 , dim( V ) − v , h ∈ V ′ with q ( v ) > q ( h ) > q ( v , h ) = 0. We may moreover assumethat h ∈ V Z .Recall the natural isomorphism so ( V h , q ) ≃ Λ V h . Let N correspond to the bivector v ∧ v under thisisomorphism. We see easily that Im( N ) = h v , v i , Im( N ) = h v i and N = 0. (cid:3) Remark . One can show that for any degeneration of K3-type Hodge structures with unipotent mono-dromy the nilpotent operator N is of the same form as in the proof above (see [SS, Proposition 4.1]). Forthe other type of degenerations (such that N = 0, N = 0) the operator N is given by w ∧ w , where h w , w i ⊂ V is an isotropic subspace. It is clear that such subspaces exist whenever dim( V ) > Definition 4.3. Let N ∈ so ( V h , q ) be nilpotent and x ∈ ˆ D h . The pair ( N, x ) defines a nilpotent orbit ifthere exists t > , such that e itN x ∈ D h for all t > t . The corresponding nilpotent orbit is the image of C in ˆ D h under the map z e zN x . Lemma 4.4. Fix N and h as in Lemma 4.1. For x ∈ ˆ D h the pair ( N, x ) defines a nilpotent orbit if andonly if q ( N x, N ¯ x ) > . The set of such points in ˆ D h is open and non-empty.Proof. The condition that should be satisfied is the following: q ( e itN x, e − itN ¯ x ) > 0, or equivalently q ( e itN x, ¯ x ) > t ≫ 0. We have e itN = 1 + 2 itN − t N , and our condition is equivalent to − q ( N x, ¯ x ) = q ( N x, N ¯ x ) > 0. The set of such x ∈ ˆ D h is clearly open. It is non-empty, because q issemi-positive on Im( N ). (cid:3) Projective degeneration. Our aim now is to prove that some of the nilpotent orbits from Lemma4.4 are induced by projective degenerations of X , i.e. obtained as the period map for such degenerations.Let us fix N and h as in Lemma 4.1, and let Γ ⊂ O( V Z , q ) be a torsion-free arithmetic subgroup. Thefollowing lemma is well-known to the experts and admits many different proofs. For completeness wesketch a proof via Hilbert schemes. Lemma 4.5. There exists a projective family ϕ : Y → S of hyperkähler manifolds deformation equivalentto X over a smooth quasi-projective base S , and a commutative diagram (4.1) ˜ S D h S D h / Γ q ˜ ρ pρ In this diagram: ˜ S is the universal covering of S , ˜ ρ is the period map for the family q ∗ Y , and ρ is anétale algebraic morphism, such that S is a finite unramified covering of ρ ( S ) .Proof. Surjectivity of the period map for hyperkähler manifolds implies that we can find a deformationof X whose period is contained in D h . Let Y be such a deformation. We can also assume that the Picardgroup of Y has rank one. We fix an isomorphism H ( Y, Z ) ≃ V Z , so that h generates Pic( Y ).The manifold Y is projective by [Hu1, Theorem 3.11] (see erratum to that paper for the correct proof),and after replacing h by its multiple we can assume that h = [ L ] for a very ample line bundle L ∈ Pic( Y ), uch that H i ( Y, L ) = 0 for all i > 0. Let W = H ( Y, L ) and consider the embedding of Y into P ( W ∗ ).This determines a C -point [ Y ] in the Hilbert scheme Hilb( P ( W ∗ ) / C ). Let H be an irreducible componentcontaining this point, and φ : Y → H the universal family. Let S ′ ⊂ H red be the maximal Zariski-opensubset over which the restriction of φ to H red is smooth; it is non-empty since it contains the point [ Y ].The family Y induces a variation of Hodge structures on the local system V = ( R φ ∗ Z ) pr with fibre V h Z over S ′ .Consider the universal covering q ′ : ˜ S ′ → S ′ . The pull-back q ′∗ V induces the period map ρ ′ : ˜ S ′ → D h and the monodromy representation µ : π ( S ′ ) → O( V h Z , q ). Consider the group Γ ′ = µ − (Γ). It is offinite index in π ( S ′ ) and we have the corresponding finite covering S ′′ . By construction, the morphism˜ ρ ′ descends to ρ ′′ : S ′′ → D h / Γ.We claim that ρ ′′ is dominant. To see this, one can consider the universal deformation of the pair( Y, L ). By the local Torelli theorem the base of this deformation is an open subset B ⊂ D h isomorphicto a polydisc. The universal family over B induces, possibly after shrinking B , a morphism B → S ′ thatlifts to B → S ′′ since B is simply-connected. The composition with ρ ′′ gives a map B → D h / Γ, that byconstruction equals the composition B ֒ → D h → D h / Γ. The image of ρ ′′ thus contains an open subset of D h / Γ, so ρ ′′ is dominant.Finally, to construct S that satisfies all the conditions of the lemma, one can take a subvariety of S ′′ that has the same dimension as D h / Γ and maps dominantly to it. Restriction of ρ ′′ to some open subsetof such subvariety will be étale. By shrinking this open subset further we can make ρ ′′ finite over itsimage. We let S be the resulting locally closed subvariety of S ′′ . The preimage q ′− ( S ) is a covering of S .We can replace it by the universal covering ˜ S , and the period map ρ then factors through q ′− ( S ). (cid:3) Theorem 4.6. Let X be a simple hyperkähler manifold with b ( X ) > . Then there exists a projective de-generation of X with maximal unipotent monodromy. The limit mixed Hodge structures on all cohomologygroups of this degeneration are of Hodge-Tate type.Proof. Consider the diagram 4.1. Let M be a smooth projective variety that contains D h / Γ as an opensubset. Let U be the image of ρ and D the complement of U inside M with reduced scheme structure.Let N be a nilpotent endomorphism from Lemma 4.1. After multiplying N by a positive integer, wemay assume that γ = e N ∈ Γ. Let N = { x ∈ D h | ( N, x ) defines a nilpotent orbit } . It follows fromLemma 4.4 and the definition of the nilpotent orbit that N is open and non-empty. Hence there exists apoint x ∈ N ∩ p − ( U ).Let us consider the nilpotent orbit given by ( N, x ). Choose t > α : { z ∈ C | Im( z ) > t } → D h , z e zN x . We have ˜ α ( z + 1) = γ ˜ α ( z ), so ˜ α descends to a map α : ∆ ∗ ε → D h / Γ, where ∆ ∗ ε is the punctured disc of radius ε = e − t . According to Borel, we can extend α over the puncture and get a map ∆ ε → M .Note that the image of α is not contained in D by the choice of x . Since α − ( D ) is an analyticsubvariety of ∆ ε , we can find ε ′ ε , such that α − ( D ) ∩ ∆ ∗ ε ′ = ∅ . We get a map α : ∆ ∗ ε ′ → U . Afterpassing to a finite unramified covering of ∆ ∗ ε ′ and rescaling the coordinate on the disc, we obtain a map α ′ : ∆ ∗ → S . We can find projective compactifications ¯ Y and ¯ S of Y and S from Lemma 4.5, such that φ extends to ¯ φ : ¯ Y → ¯ S . According to Borel and Kobayashi, α ′ extends to a morphism ∆ → ¯ S . Then thepull-back of ¯ Y by α ′ defines a projective degeneration over ∆. The assertion about the limit mixed Hodgestructures follows from Theorem 3.8. (cid:3) cknowledgements I am grateful to Misha Verbitsky and Daniel Hyubrechts for useful discussions and the reference [To],and to the referee for valuable comments and suggestions. References [Bor] A. Borel, Introduction aux groupes arithmétiques , Publications de l’Institut de Mathématique de l’Université deStrasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.[Bou] N. Bourbaki, Lie groups and Lie algebras. Chapters 7–9 , Springer-Verlag, Berlin, 2005, xii+434 pp.[De] P. 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Verbitsky, Cohomology of compact hyperkähler manifolds , Ph.D. dissertation, Harvard University, Cambridge,Mass., 1995; arXiv:alg-geom/9501001[Ve2] M. Verbitsky, Mirror Symmetry for hyper-Kähler manifolds , in Mirror Symmetry, III (Montreal, 1995), AMS/IPStud. Adv. Math. 10, Amer. Math. Soc., Providence, 1999, 115–156.[Ve3] M. Verbitsky, Mapping class group and a global Torelli theorem for hyperkähler manifolds , Duke Math. J. 162(2013), 2929–2986. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin E-mail address : [email protected]@hu-berlin.de