k-symplectic structures and absolutely trianalytic subvarieties in hyperkahler manifolds
aa r X i v : . [ m a t h . AG ] F e b A. Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties k -symplectic structures andabsolutely trianalytic subvarieties inhyperk¨ahler manifolds. Andrey Soldatenkov , Misha Verbitsky Abstract
Let (
M, I, J, K ) be a hyperk¨ahler manifold, and Z ⊂ ( M, I ) a complexsubvariety in (
M, I ). We say that Z is trianalytic if it is complexanalytic with respect to J and K , and absolutely trianalytic if it istrianalytic with respect to any hyperk¨ahler triple of complex structures( M, I, J ′ , K ′ ) containing I . For a generic complex structure I on M , allcomplex subvarieties of ( M, I ) are absolutely trianalytic. It is knownthat the normalization Z ′ of a trianalytic subvariety is smooth; weprove that b ( Z ′ ) > b ( M ), when M has maximal holonomy (that is, M is IHS).To study absolutely trianalytic subvarieties further, we define anew geometric structure, called k -symplectic structure; this structureis a generalization of hypersymplectic structure. A k -symplectic struc-ture on a 2 d -dimensional manifold X is a k -dimensional space R ofclosed 2-forms on X which all have rank 2 d or d . It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R .We consider absolutely trianalytic tori in a hyperk¨ahler manifold M of maximal holonomy. We prove that any such torus is equippedwith a non-degenerate k -symplectic structure, where k = b ( M ). Weshow that the tangent bundle T X of a k -symplectic manifold is aClifford module over a Clifford algebra Cl ( k − M with b ( M ) > r + 1 isat least 2 r − -dimensional. Contents Andrey Soldatenkov is partially supported by AG Laboratory NRU-HSE, RF gov-ernment grant, ag. 11.G34.31.0023, MK-1297.2014.1 and a grant from Dmitri Zimin’s“Dynasty” foundation Partially supported by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE. – 1 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties k -symplectic structures and absolutely triana-lytic subvarieties . . . . . . . . . . . . . . . . . . . . . . . . . 7 k -symplectic structures and Clifford representations 9 k -symplectic structures on vector spaces . . . . . . . . . . . . 92.2 Applications to hyperk¨ahler geometry . . . . . . . . . . . . . . 15 k -symplectic structures on manifolds . . . . . . . . . . . . . . 173.2 Open questions and possible directions of research . . . . . . . 19 Let M be a K¨ahler, compact, holomorphic symplectic manifold. Calabi-Yautheorem ([Y]) implies that M admits a Ricci-flat metric g , unique in eachK¨ahler class. Using Berger’s classification of Riemannian holonomies andBochner vanishing, one shows that the Levi-Civita connection of g preserves atriple of complex structures I, J, K satisfying the quaternionic relation IJ = − J I = K ([Bes]). A Riemannian manifold admitting a triple of complexstructures I, J, K satisfying quaternionic relations and K¨ahler with respectto g is called hyperk¨ahler . One can construct a holomorphic symplecticform on any hyperk¨ahler manifold as follows. There are K¨ahler forms ω I , ω J and ω K associated to the complex structures I, J and K . One can check thatthe 2-form ω J + √− ω K is of Hodge type (2 ,
0) with respect to I . Since it isalso closed it is a holomorphic symplectic form. So we shall treat the terms“hyperk¨ahler” and “holomorphic symplectic” as (essentially) synonyms.Given any triple a, b, c ∈ R , a + b + c = 1, the operator L = aI + bJ + cK satisfies L = − M, g ); such a complexstructure is called induced by the hyperk¨ahler structure . Complexsubvarieties of such (
M, L ) for generic ( a, b, c ) were studied in [V1], [V2] and[V3].
Definition 1.1:
Let Z ⊂ M be a closed subset of a hyperk¨ahler manifold.It is called trianalytic , if it is complex analytic with respect to all inducedcomplex structures L . – 2 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties In the definition above it is enough to require the subvariety Z to becomplex analytic with respect only to I and J . Then it will automaticallybe complex analytic with respect to any induced complex structure. This isclear, because Z is trianalytic if and only if for all smooth points z ∈ Z , thespace T z Z ⊂ T z M is preserved by the quaternion algebra H ([V2]). However, H is generated by any two non-collinear elements I, I with I = I = − Theorem 1.2: ([V5]) Let M be a hyperk¨ahler manifold, Z ⊂ M a trianalyticsubvariety, and I an induced complex structure. Consider the normalization ^ ( Z, I ) n −→ ( Z, I )of (
Z, I ). Then ^ ( Z, I ) is smooth, and the map ^ ( Z, I ) −→ M is an immersion,inducing a hyperk¨ahler structure on ^ ( Z, I ). Definition 1.3:
Let M be a hyperk¨ahler manifold, and S the family of allinduced complex structures L = aI + bJ + cK , where a, b, c ∈ R , a + b + c =1. Then S is called the twistor family of complex structures.The following theorem implies that whenever L is a generic element of atwistor family, all subvarieties of ( M, L ) are trianalytic.
Theorem 1.4: ([V2, V3]) Let M be a hyperk¨ahler manifold, S its twistorfamily. Then there exists a countable subset S ⊂ S , such that for anycomplex structure L ∈ S \ S , all compact complex subvarieties of ( M, L ) aretrianalytic.In [V6] (see also [V7], [KV2] and [O]), this theorem was used to studysubvarieties of generic deformations of a compact holomorphic symplecticmanifold M . Recall that the Teichm¨uller space of M is the quotient Teich = Comp / Diff of the (infinite-dimensional) space of all complex structures ofhyperk¨ahler type by the group Diff of isotopies ([V9]). Teich is a complex,non-Hausdorff manifold. If we fix a complex structure I on M then theconnected component of the Teichm¨uller space containing I can be identi-fied with a connected component of the so-called “marked moduli space” ofdeformations of ( M, I ) ([V9]). – 3 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties Definition 1.5:
Let (
M, I, J, K ) be a compact, holomorphic symplectic,K¨ahler manifold, and Z ⊂ ( M, I ) a complex subvariety, which is trianalyticwith respect to any hyperk¨ahler structure compatible with I . Then Z iscalled absolutely trianalytic . Definition 1.6:
For a given complex structure I , consider the Weil oper-ator W I acting on ( p, q ) forms as √− p − q ). Let G MT ( M, I ) be a smallestrational algebraic subgroup of Aut( H ∗ ( M, R )) containing e tW I . This groupis called the Mumford-Tate group of ( M, I ). A group generated by G MT ( M, I ) for all complex structures I in a connected component of a de-formation space is called a maximal Mumford-Tate group of M ([D]).It is not hard to check that the Mumford-Tate group G MT ( I ) of ( M, I )is lower semicontinuous as a function of I ⊂ Teich in Zariski topology on
Teich ([D]). This implies that G MT ( I ) is constant outside of countably manycomplex subvarieties of positive codimension. We call I ∈ Teich
Mumford-Tate generic if G MT ( I ) is maximal. If M has maximal holonomy, the max-imal Mumford-Tate group is isomorphic to Spin( H ( M, R ) , q ) ([V7]). Any I ∈ Teich outside of a countably many subvarieties of positive codimensionis Mumford-Tate generic.
Remark 1.7:
Let I be a Mumford-Tate generic complex structure, and η an integer ( p, p )-class. Then η is of type ( p, p ) for any deformation of I .Absolutely trianalytic subvarieties can be characterized in terms of theMumford-Tate group, as follows. Claim 1.8:
Let (
M, I, J, K ) be a hyperk¨ahler manifold, and Z ⊂ ( M, I )be a complex subvariety. Then Z is absolutely trianalytic if and only if itsfundamental class is G -invariant, where G is a maximal Mumford-Tate groupof M . In particular, Z is absolutely trianalytic when ( M, I ) is Mumford-Tategeneric.
Proof:
This statement follows from the definitions ([V8, Claim 4.4]).Clearly, the set of absolutely trianalytic subvarieties does not change ifone passes from one complex structure in a twistor family S to anothercomplex structure in S . On the other hand, any two complex structures– 4 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties in the same component of Teich can be connected by a sequence of twistorfamilies ([V4]). This means that the set of absolutely trianalytic subvarietiesin (
M, I ) is determined by the connected component of
Teich where I lies.This can be used to prove the following theorem. Theorem 1.9:
Let I , I ∈ Teich be points in the same connected com-ponent of the Teichm¨uller space. Then there exists a diffeomorphism ν :( M, I ) −→ ( M, I ) such that any absolutely trianalytic subvariety Z ⊂ ( M, I )is mapped to an absolutely trianalytic subvariety ν ( Z ) ⊂ ( M, I ).Absolutely trianalytic subvarieties were studied in [V6], where it wasshown that a general deformation of a Hilbert scheme of a K3 surface has nocomplex (or, equivalently, no absolutely trianalytic) subvarieties. In [KV1],Kaledin and Verbitsky used the same argument to study absolutely tri-analytic subvarieties in generalized Kummer varieties. They “proved” non-existence of such subvarieties, but their argument was faulty (in fact, thereexists an absolutely trianalytic subvariety in a generalized Kummer variety).In [KV2], the error was found, and the argument was repaired to show thatany absolutely trianalytic subvariety Z of a generalized Kummer variety isa deformation of a resolution of singularities of a quotient of a torus by aWeyl group action. The idea was to show that Z is a resolution of singular-ities of a quotient of a flat subtorus in a symmetric power of a torus, andclassify the group actions which admit a holomorphic symplectic resolution.Later, Ginzburg and Kaledin have shown that only the Weyl groups A n , B n , C n can occur in quotient maps with quotients which admit holomorphicallysymplectic resolution of singularities ([GK]).Non-existence of absolutely trianalytic subvarieties in a Hilbert scheme M of K3 was used in [V7] to prove compactness of deformation spaces ofcertain stable holomorphic bundles on M . Before we state our main results, let us introduce the Bogomolov-Beauville-Fujiki form and maximal holonomy manifolds.
Definition 1.10:
A compact hyperk¨ahler manifold M is called simple , or maximal holonomy , or IHS (from “Irreducible Holomorphic Symplectic”)if π ( M ) = 0 and H , ( M ) = C . Remark 1.11:
It follows from Bochner’s vanishing and Berger’s classifica-– 5 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties tion of holonomy groups that a hyperk¨ahler manifold has maximal holonomySp( n ) whenever π ( M ) = 0, H , ( M ) = C ([Bes]). This explains the term. Theorem 1.12: (Bogomolov’s decomposition; [Bo1]) Any compact hyperk¨ahlermanifold admits a finite covering which is a product of a torus and severalsimple hyperk¨ahler manifolds.
Theorem 1.13: (Fujiki, [F]) Let M be a simple hyperk¨ahler manifold ofcomplex dimension 2 n . Then there exists a primitive integral quadratic form q on H ( M, Z ) and a constant c M such that for any η ∈ H ( M, C ) we have R M η n = c M q ( η, η ) n . Remark 1.14:
Theorem 1.13 determines the form q uniquely up to a sign.For dim R M = 4 n , n odd, sign is also determined. For n even, the sign isdetermined by the following formula, due to Bogomolov and Beauville. µq ( η, η ) = ( n/ Z X η ∧ η ∧ Ω n − ∧ Ω n − −− (1 − n ) (cid:18)Z X η ∧ Ω n − ∧ Ω n (cid:19) (cid:18)Z X η ∧ Ω n ∧ Ω n − (cid:19) (1.1)where Ω is the holomorphic symplectic form, and µ > Definition 1.15:
Let M be a hyperk¨ahler manifold of maximal holonomy,and q the form on H ( M ) defined by Remark 1.14 and Theorem 1.13. Then q is called Bogomolov-Beauville-Fujiki form (BBF form). The equation(1.1) (or, even better, a similar equation [V0, (1.1)], using the K¨ahler forminstead of the holomorphic symplectic form) can be used to show that q isnon-degenerate and has signature (3 , b ( M ) − M be a simple hyperk¨ahlermanifold with dim C M = 2 n . Consider a cohomology class γ which is ofHodge type (2 n − m, n − m ) on any small deformation of M . Then thereexists a constant c γ ∈ R such that for any α, β ∈ H ( X, C ) we have γ · α m − · β = c γ q ( α, α ) m − q ( α, β ) . (1.2)In this equation on the left hand side we have the intersection product incohomology of M . – 6 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties k -symplectic structures and abso-lutely trianalytic subvarieties In the present paper, we prove two bounds on the Betti numbers of absolutelytrianalytic subvarieties, quite restrictive for their geometry. In particular, weprove that the normalization of a proper complex subvariety of a genericdeformation of a 10-dimensional O’Grady space must necessarily belong toa new type of hyperk¨ahler manifolds (Theorem 1.20).Our arguments are based on an elementary observation, stated below asTheorem 1.16.Recall (Theorem 1.2) that the normalization of any trianalytic subvariety
Z ֒ → M is a smooth hyperk¨ahler manifold ˜ Z immersed into M . Therefore,we can replace any trianalytic cycle by an immersed hyperk¨ahler manifold.Note that the complex dimension of any trianalytic subvariety is even. Theorem 1.16:
Let M be a maximal holonomy hyperk¨ahler manifold and˜ Z ϕ −→ M the normalization of an absolutely trianalytic cycle Z ⊂ M or a fi-nite covering of such normalization. Consider the polynomial P ˜ Z on H ( ˜ Z, C )mapping a cohomology class η to R ˜ Z η dim C ˜ Z . Then for any α ∈ H ( M, C ), onehas P ˜ Z ( ϕ ∗ α ) = deg( ϕ ) c Z q ( α, α ) dim C Z , where c Z is a constant determinedby M and Z . Proof:
Denote by γ the fundamental class of Z in the cohomology of M .We have P ˜ Z ( ϕ ∗ α ) = deg( ϕ ) γ · α dim C Z = c Z q ( α, α ) dim C Z . The last equalityfollows from (1.2) where we put β = α and c Z = c γ .This simple result has very nice consequences. Corollary 1.17:
Let M be a maximal holonomy hyperk¨ahler manifold, and˜ Z ϕ −→ M the normalization of an absolutely trianalytic cycle Z ⊂ M , or afinite covering of the normalization. Then the induced map H ( M, C ) ϕ ∗ −→ H ( ˜ Z, C ) is injective. Proof:
Given x ∈ ker ϕ ∗ ⊂ H ( M, C ) and any y ∈ H ( M, C ), one wouldobtaindeg( ϕ ) c Z q ( x + y, x + y ) dim C Z = P ˜ Z ( ϕ ∗ ( x + y )) = P ˜ Z ( ϕ ∗ y ) == deg( ϕ ) c Z q ( y, y ) dim C Z . – 7 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties This gives q ( x, y ) = 0 for all y ∈ H ( M, C ). However, this implies x = 0,because q is non-degenerate. Corollary 1.18:
Let M be a maximal holonomy hyperk¨ahler manifold, Z ⊂ M an absolutely trianalytic subvariety, and ˜ Z −→ M its normalization.Consider a finite covering ˜ Z −→ ˜ Z such that ˜ Z is a product of a hyperk¨ahlertorus T and several maximal holonomy hyperk¨ahler manifolds K i (such adecomposition always exists by Bogomolov decomposition Theorem 1.12).Then b ( T ) > b ( M ) and b ( K i ) > b ( M ). Proof:
Any hyperk¨ahler metric on M induces a hyperk¨ahler structureon ˜ Z . Each component in Bogomolov decomposition of ˜ Z is immersed into M as a hyperk¨ahler subvariety. Hence the images of those components arealso absolutely trianalytic subvarieties of M and we can apply Corollary 1.17to them.In this paper we are also interested in absolutely trianalytic subvarieties Z ⊂ M such that the normalization ˜ Z is a torus. We prove the followingtheorem. Theorem 1.19:
Let M be a hyperk¨ahler manifold of maximal holonomy, T a hyperk¨ahler torus, and T −→ M a hyperk¨ahler immersion with absolutelytrianalytic image. Then b ( T ) > ⌊ ( b ( X ) − / ⌋ . Proof:
Corollary 2.15.Together with Corollary 1.18, this allows to prove non-existence of subva-rieties of known type in a 10-dimensional O’Grady manifold M (Remark 2.16),which has b ( M ) = 24. Theorem 1.20:
Let Z ⊂ M be a proper complex subvariety of a gen-eral deformation of 10-dimensional O’Grady manifold, ˜ Z its normalization,and ˜ Z its covering equipped with the Bogomolov decomposition obtainedas in Corollary 1.18. Then ˜ Z i = Q i K i where K i are maximal holonomyhyperk¨ahler manifolds with b >
24, that is, previously unknown type.
Remark 1.21:
A maximal holonomy hyperk¨ahler manifold K i with dim C K i =2 satisfies b ( K i )
22 (Kodaira-Enriques classification, see e.g. [Bes]). Whendim C K i = 4, one has b ( K i )
23 ([G]). Therefore, any absolutely triana-lytic subvariety in a 10-dimensional O’Grady manifold (if it exists) satisfies– 8 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties dim C Z >
6, and has maximal holonomy.For a 6-dimensional O’Grady manifold M , one has b ( M ) = 8, so we cannot prove analogous result by our methods. However, complex tori are stillprohibited. Theorem 1.22:
Let M be a general deformation of a 6-dimensional O’Gradymanifold. Then any holomorphic map from a complex torus to M is trivial. Proof:
Since an image of such a map is absolutely trianalytic, its nor-malization is hyperk¨ahler. However, a dominant holomorphic map from atorus T to a manifold Z with trivial canonical bundle is a projection to aquotient torus. This is clear because fibers of such map have trivial canonicalbundle by adjunction formula, but a Calabi-Yau submanifold in a torus isalso a torus. Then Theorem 1.22 follows from Remark 2.17. k -symplectic structures and Clifford repre-sentations The proof of Theorem 1.19 is based on a discovery of a previously unknowngeometric structure, called k -symplectic structure. In Subsection 2.1 westudy k -symplectic structures on vector spaces, and in Subsection 2.2 weapply these linear-algebraic results to algebraic geometry of absolutely tri-analytic tori. k -symplectic structures generalize the hypersymplectic structures knownfor a long time ([Ar], [DS1], [AD]), and trisymplectic (3-symplectic) struc-tures defined in [JV2]. k -symplectic structures on vector spaces Consider a complex vector space V of dimension dim C V = 4 n . Let k be anon-negative integer. Definition 2.1: A k -symplectic structure on V is a subspace Ω ⊂ Λ V ∗ of dimension k , such that for some non-zero quadratic form q ∈ S Ω ∗ thefollowing condition is satisfied: for any non-zero ω ∈ Ω we havedim(ker ω ) = ( n, if q ( ω ) = 0;0 , otherwise.– 9 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties A k -symplectic structure is called non-degenerate if the quadratic form q isnon-degenerate. A vector space with a k -symplectic structure will be calleda k -symplectic vector space.A few remarks about this definition. Remark 2.2:
The quadratic form q in the definition above is unique up toa non-zero multiplier. We can reformulate the condition from the definitionas follows. Consider the Pfaffian variety P = { ω ∈ P (Λ V ∗ ) | ω n = 0 } in P (Λ V ∗ ). A subspace Ω ⊂ Λ V ∗ defines a k -symplectic structure if andonly if P Ω ∩ P is a quadric (necessarily of multiplicity n ), and all the forms ω lying on this quadric have rank 2 n . The quadratic form q is the one definingthis quadric. If it is necessary to mention the form q explicitly, we will denotea k -symplectic structure by ( Ω , q ). Remark 2.3:
A 1-symplectic structure is the same thing as a non-zero two-form ω ∈ Λ V ∗ which is either non-degenerate or has rank 2 n . Remark 2.4:
Consider a non-degenerate 2-symplectic structure ( Ω , q ) on V . The quadric in P Ω = P defined by q is non-degenerate, hence it consistsof two distinct points. Denote the corresponding two-forms by ω and ω .The kernels V and V of these forms have dimension 2 n by definition ofa 2-symplectic structure. Moreover, V ∩ V = 0, because the generic linearcombination of ω and ω must be non-degenerate. Then we have V = V ⊕ V .Conversely, given a symplectic vector space ( W, ω ), consider the directsum V = W ⊕ and denote by π i : V → W the projection to the i -th sum-mand. Then the subspace Ω ⊂ Λ V ∗ spanned by π ∗ ω and π ∗ ω defines anon-degenerate 2-symplectic structure on V . Remark 2.5:
Given a k -symplectic structure ( Ω , q ), we can consider anysubspace Ω ′ ⊂ Ω . Then ( Ω ′ , q | Ω ′ ) is a k ′ -symplectic structure for k ′ =dim C Ω ′ . We will call it a substructure of ( Ω , q ). Note that such substructurecan be degenerate even if the initial structure was not. Remark 2.6:
Consider a vector space V of complex dimension 4. Then Ω = Λ V gives a 6-symplectic structure on V , since the set of all degeneratetwo-forms on V is a Pl¨ucker quadric, and all non-zero degenerate two-formshave two-dimensional kernel. – 10 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties There exist examples of non-degenerate k -symplectic structures for every k , as follows from the construction described below.Recall the definition of Clifford algebras (for the proofs of all statementsabout Clifford algebras, see [ABS]). We will always assume that all vectorspaces are either real or complex, however the definition below is valid for anyfield. Consider a vector space E with quadratic form q ∈ S E ∗ . By definition,the Clifford algebra C l( E, q ) is the quotient of the tensor algebra T • E by thetwo-sided ideal generated by all tensors of the form x ⊗ x − q ( x, x ) ·
1, forall x ∈ E . The elements generating the ideal belong to the even part of thetensor algebra, so the Clifford algebra is Z / Z -graded, C l( E, q ) = C l ( E, q ) ⊕C l ( E, q ). There exists an automorphism τ : C l( E, q ) → C l( E, q ), which actsas the identity on C l ( E, q ) and as multiplication by − C l ( E, q ). Thereis also an antiautomorphism of transposition which maps x = x · . . . · x k to x t = x k · . . . · x for any x i ∈ E . We will use the notation x = τ ( x t ) for x ∈ C l( E, q ).One can check that the space E is embedded into C l( E, q ) via the map E = T E → C l( E, q ) induced by the projection T • E → C l( E, q ). TheClifford algebra has the following universal property. Let A be any associativealgebra with unit and α : E → A a map with the property that α ( x ) = q ( x, x ) · A for all x ∈ E . Then α can be uniquely extended to a morphismof algebras α ′ : C l( E, q ) → A such that α ′ | E = α .Recall the definition of the group Pin( E, q ) (see [ABS]). We will denoteby C l × ( E, q ) the multiplicative group of invertible elements. DefinePin(
E, q ) = { x ∈ C l × ( E, q ) | τ ( x ) Ex − = E, xx = 1 } . One can check that Pin(
E, q ) acts on E preserving the quadratic form q andis actually a double covering of the orthogonal group O( E, q ). The groupSpin(
E, q ) is defined as the subgroup of even elements in Pin(
E, q ). Example 2.7:
For each integer k > k -symplectic structure on some vector space. Start from a real k -dimensionalvector space E R with a negative-definite quadratic form q . Consider the Clif-ford algebra C l( E R , q ) and the corresponding group Pin( E R , q ). This groupis compact (because q is negative-definite) and it acts on E R by orthogonaltransformations. Lemma 2.8:
Consider a non-trivial real representation ρ : C l( E R , q ) → End( V R )in some real vector space V R , dim V R = 4 n (by this we mean a representation– 11 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties of C l( E R , q ) as an algebra with unit). For any representation V R of C l( E R , q )as above, we have a Spin( E R , q )-equivariant embedding α R : E R ֒ → Λ V ∗ R . All non-zero two-forms in the image of α R are non-degenerate. Proof:
If we have a representation V R , then the compact group Pin( E R , q )acts on V R , and there exists a positive-definite invariant quadratic form g ∈ ( S V ∗ R ) Pin( E R ,q ) .Note that any element e ∈ E R with e = − E R , q ). Thisfollows from direct computation: we have τ ( e ) = − e , ee = 1 and for any x ∈ E R we have τ ( e ) xe − = exe = 2 q ( x, e ) e − xe = 2 q ( x, e ) e + x ∈ E R .From this and the invariance of g with respect to Pin( E R , q ) we concludethat g ( e · u, e · v ) = g ( u, v ) for any u, v ∈ V R .We can define the two-form ω e ∈ Λ V ∗ R by the formula ω e ( u, v ) = g ( e · u, v )for u, v ∈ V R . This really defines an element in Λ V ∗ R since g ( e · u, v ) = g ( e · u, e · v ) = − g ( u, e · v ) = − g ( e · v, u ).The quadratic form q was chosen negative-definite, so every element x ∈ E R is of the form x = ξe with ξ ∈ R and e = −
1, so ω x = ξω e ∈ Λ V ∗ R .We have constructed a map α R : E R ֒ → Λ V ∗ R , x ω x , which is clearly anembedding.Note that this embedding is a morphism of Spin( E R , q )-modules: for any h ∈ Spin( E R , q ) we have τ ( h ) = h , α R ( h · x ) = α R ( hxh − ) = ω hxh − and ω hxh − ( u, v ) = g ( hxh − · u, v ) = g ( xh − · u, h − · v ) = ω x ( h − · u, h − · v ), so α R ( h · x ) = h · ω x .For all non-zero x ∈ E R the form ω x is non-degenerate, because thequadratic form g is chosen positive-definite (and in particular non-degenerate).Consider the complexifications: E = E R ⊗ C , C l( E, q ) = C l( E R , q ) ⊗ C and V = V R ⊗ C with the corresponding complex representation of C l( E, q ).Then we get an embedding of complex vector spaces α : E ֒ → Λ V ∗ .So we have an irreducible Spin( E, q )-module E embedded into Λ V ∗ . Fixa volume form and identify Λ n V ∗ with C . Then the wedge product in Λ • V ∗ induces a Spin( E, q )-invariant polynomial p ∈ S n E ∗ , p ( x ) = ω ∧ nx on E .This polynomial is non-zero, because the image of α contains non-degeneratetwo-forms by Lemma 2.8.The group Spin( E, q ) is a double covering of SO(
E, q ), so p is SO( E, q )-invariant and by classical invariant theory this polynomial has to be propor-tional to the n -th power of q . – 12 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties We can define Ω to be the image of E in Λ V ∗ under the map α . We havealready seen that the set of degenerate two-forms is a quadric. To see that Ω is a k -symplectic structure it only remains to check that all degeneratetwo-forms in Ω have kernels of dimension 2 n . This is proved in the followinglemma which we will also need later on. Lemma 2.9:
Let V be a complex vector space of dimension 4 n . Let Ω ⊂ Λ V ∗ be a k -dimensional subspace. Identifying Λ n C V ∗ with C , consider apolynomial p ∈ S n Ω ∗ given by p ( ω ) = ω ∧ n . Suppose that there existsa non-degenerate quadratic form q ∈ S Ω ∗ , such that p = q n . Then alldegenerate two-forms in Ω have rank 2 n , in particular Ω is a k -symplecticstructure on V . Proof:
Let ω ∈ Ω be such that q ( ω , ω ) = 0, and let dim(ker ω ) =2 r . Since q is non-degenerate we can find ω ∈ Ω with q ( ω , ω ) = 0 and q ( ω , ω ) = 0. Consider the polynomial ˜ p ( t ) = p ( ω + tω ). We have ˜ p ( t ) = q ( ω + tω , ω + tω ) n , so ˜ p must have zero of order n at t = 0. But˜ p ( t ) = ( ω + tω ) ∧ n = t r ω ∧ (2 n − r )1 ∧ ω ∧ r + t r +1 ω ∧ (2 n − r − ∧ ω ∧ ( r +1)2 + . . . + t n ω ∧ n , and ω ∧ (2 n − r )1 ∧ ω ∧ r = 0 because ω is non-degenerate. So the order of zero at t = 0 is r , hence r = n .We can bound from below the dimension of a vector space carrying a k -symplectic structure. In the proposition below we use the following termi-nology: Definition 2.10: A k -symplectic structure ( Ω , q ) on a vector space V iscalled real if V is a complexification of a real vector space V R , Ω is a com-plexification of a real subspace Ω R in Λ V ∗ R , and q is a real quadratic form.In particular, the k -symplectic structures from the previous example arereal. We will denote by C l r,s the Clifford algebra for a real ( r + s )-dimensionalvector space with quadratic form of signature ( r, s ), meaning that it has r minuses and s pluses. Proposition 2.11: (a) Let ( V, Ω , q ) be a vector space equipped with a non-degenerate q -symplectic structure. Then V is a C l ( Ω , q )-module.– 13 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties (b) Let ( V, Ω , q ) be a vector space with a non-degenerate real k -symplecticstructure where the real quadratic form q has signature ( r, s ). Then thecorresponding real vector space V R has a structure of C l r − ,s -module if r >
0, and of a C l s − ,r -module if s > Proof:
Consider a pair of elements ω , ω ∈ Ω , such that q ( ω , ω ) = − q ( ω , ω ) = 0. These elements define linear maps ω i : V → V ∗ , and bydefinition of a k -symplectic structure the map ω is an isomorphism. So wecan define the endomorphism A = ω − ω ∈ End( V ). We claim that A = q ( ω , ω ) Id. (2.1)To prove this we need to find the eigenvalues of A : the operator A − λId = ω − ( ω − λω ) is degenerate if and only if the form ω − λω has non-trivialkernel. By definition of a k -symplectic structure, this condition is equivalentto q ( ω − λω , ω − λω ) = 0, hence the eigenvalues of A are λ ± = ± q ( ω , ω ) / and the claim follows.Next fix an element ω ∈ Ω with q ( ω , ω ) = − q -orthogonal complement W = { ω ∈ Ω | q ( ω , ω ) = 0 } . Define a linear map α : W → End( V ), α ( ω ) = ω − ω . From the equation (2.1), we see that α ( ω ) − q ( ω , ω ) Id = 0. By the universal property of the Clifford algebra,the map α can be extended to α ′ : C l( W, q | W ) → End( V ).Observe that C l( W, q | W ) ≃ C l ( Ω , q ). This isomorphism can be con-structed as follows. For any element η ∈ C l( W, q | W ) consider the decompo-sition η = η + η into even and odd parts. Then the map η η + ω η ∈C l ( Ω , q ) is the desired isomorphism, which can be checked using the factthat ω = − ω commutes with even and anticommutes with oddelements from C l( W, q | W ). This proves the first part of the proposition.Next consider the case of a real k -symplectic structure with the quadraticform q of signature ( r, s ). If r > ω ∈ Ω R with q ( ω , ω ) = −
1. Its real q -orthogonal complement W R has a quadraticform of signature ( r − , s ), and we obtain a representation of C l r − ,s in V R .If s > ω ∈ Ω R with q ( ω , ω ) = 1. Inthis case for any ω with q ( ω , ω ) = 0 we have an operator A = ω − ω with A = − q ( ω , ω ) Id . Then we obtain a representation of C l( W R , q ′ ) in V R ,where W R is the q -orthogonal complement to ω as above, and q ′ = − q | W R .The form q ′ has signature ( s − , r ), so the second claim of the propositionfollows. Corollary 2.12:
Let ( V, Ω , q ) be vector space equipped with a non-degegenerate– 14 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties k -symplectic structure. Thendim C V = 2 ⌊ ( k − / ⌋ m for some positive integer m . Proof:
By the previous proposition, V is a non-trivial C l ( Ω , q )-module,so we have a map α ′ : C l ( Ω , q ) → End( V ). The map α ′ is non-zero, so itsimage is isomorphic to a quotient of C l ( Ω , q ) by a proper two-sided ideal.But C l ( Ω , q ) is either the matrix algebra Mat(2 ( k − / , C ) if k is odd or thesum of two copies of the matrix algebra Mat(2 k/ − , C ) if k is even. In anycase, V is a direct sum of m copies of the standard representation of thematrix algebra for some m . Hence the desired equality for dim V . Remark 2.13:
Both the construction from the previous example and theproof of the previous proposition involve Clifford algebras. Note, however,that these two constructions are not inverse to each other. Not every k -symplectic structure arises from a real representation of a Clifford algebraassociated with a k -dimensional vector space. One example when this is notthe case was already mentioned above: this is a 6-symplectic structure on a4-dimensional vector space V . In this case, the proof of the previous propo-sition gives us a representation α ′ : C l( W, q ) → End( V ) with 5-dimensionalvector space W and C l( W, q ) isomorphic to Mat(4 , C ) ⊕ Mat(4 , C ). The map α ′ here is not injective. Note that by dimension reasons V cannot be a rep-resentation of a Clifford algebra associated to a 6-dimensional vector space.If we consider V as a complexification of a real 4-dimensional vector space,then the corresponding quadratic form on Λ V ∗ R will be of signature (3 , C l , -module structure on V R . Thealgebra C l , is isomorphic to Mat(4 , R ) ⊕ Mat(4 , R ). Next we will use our observations about k -symplectic structures to investi-gate submanifolds in a very general irreducible holomorphic symplectic (IHS)manifold (see Definition 1.10).Consider an IHS manifold X of dimension 2 n with symplectic form σ .We will call X very general if it represents a point in the moduli space whichlies in the complement to a countable union of proper analytic subvarieties.One can prove that a very general X cannot contain submanifolds of odddimension. Moreover, if there exists a submanifold of dimension 2 m inside– 15 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties such X , and γ ∈ H n − m ) ( X, Z ) is its fundamental class, then γ stays ofHodge type (2 n − m, n − m ) on any small deformation of X (such asubvariety is called absolutely trianalytic , see Definition 1.5). The proofof these well-known facts can be found for example in [GHJ], section 26.3. Proposition 2.14:
Let X be an IHS manifold, k = b ( X ) its second Bettinumber, and T a compact complex torus of dimension 2 m immersed into X .Assume that T is absolutely trianalytic (this would follow if X is sufficientlygeneral). Then H ( T, C ) carries a non-degenerate real k -symplectic structure.The corresponding quadratic form has signature ( k − , Proof:
Denote by j : T → X the immersion. Let V = H ( T, C ), thenΛ V ∗ = H ( T, C ) and we have the restriction map j ∗ : H ( X, C ) → H ( T, C ).Fujiki relations (1.2) imply that j ∗ is injective: for any α, β ∈ H ( X, C ) wehave ( j ∗ α ) m − · j ∗ β = γ · α m − · β = c γ q ( α, α ) m − q ( α, β ) , and if j ∗ β = 0 then we must have q ( α, β ) = 0 for any α with q ( α, α ) = 0.Taking α to be a K¨ahler form we see that β = 0.We define Ω = Im( j ∗ ) ⊂ Λ V ∗ . Then by Fujiki relations (1.2) we have( j ∗ α ) m = c γ q ( α, α ) m and by Lemma 2.9 Ω is a k -symplectic structure on V .This k -symplectic structure is real, because the Beauville-Bogomolov formis real (the real structure on V is the standard real structure on homology).The signature of Beauville-Bogomolov form is known to be ( b ( X ) − , Corollary 2.15:
If a torus T is immersed into a very general IHS manifold X then dim C T > ⌊ ( b ( X ) − / ⌋− . Proof:
This follows directly from Proposition 2.14 and Corollary 2.12(note that the complex dimension of a torus is twice its first Betti number).
Remark 2.16:
In particular, this shows that a very general deformation ofa 10-dimensional IHS manifold M constructed by O’Grady can not containa complex torus, because it is known that in this case b ( M ) = 24, and thiswould imply dim C T > , which is impossible. Remark 2.17:
For the 6-dimensional IHS manifold of O’Grady’s, we have b = 8 and our estimate gives dim C T >
4. But in this case we can use the– 16 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties fact that the real homology H ( T, R ) must carry a real 8-symplectic structurewith quadratic form of signature (5 , H ( T, R ) has to be a C l , -module (and a C l , -module). The algebras C l , and C l , are isomorphic toMat(8 , C ) as real algebras. The minimal non-trivial real representation ofMat(8 , C ) is C ≃ R . This means that H ( T, R ) has to be at least 16-dimensional which gives a contradiction. k -symplectic structures on manifolds The notion of k -symplectic structure generalizes that of a trisymplectic struc-ture, which was originally defined in [JV2]. Jardim and Verbitsky define atrisymplectic structure on a complex manifold M to be a triple h Ω , Ω , Ω i of symplectic forms satisfying the following condition. Let Ω be the vectorspace generated by h Ω , Ω , Ω i , and Q ⊂ Ω the set of degenerate forms.Then Q is a non-degenerate quadric hypersurface, all v ∈ Q have constantrank, and all non-zero v ∈ Q have rank dim M .This notion was studied in [JV2] when M is a complex manifold, andΩ , Ω , Ω holomorphic symplectic forms. In this case M admits an actionof Mat(2 , C ) in its tangent space, preserving the space Ω , and a torsion-freeholomorphic connection preserving the Ω and Mat(2 , C )-action. Trisymplec-tic structures arise naturally in connection with the mathematical instantonson C P ; indeed, the moduli space of framed mathematical instantons on C P is trisymplectic.Trisymplectic structure is a special case of the 3-web, explored in [JV1],where it was studied using the geometric approach coming back to Chern’sdoctoral thesis [C]. A 3-web is a triple of involutive sub-bundles S , S , S ⊂ T M such that
T M = S i ⊕ S j for any i = j . Chern has constructed a canonicalconnection on a manifold equipped with a 3-web; for 3-webs arising fromtrisymplectic structures, this connection turns out to be torsion-free.Trisymplectic manifold can be characterized in terms of holonomy of thisconnection. These are manifolds equipped with a torsion free connection ∇ such that its holonomy group Hol( ∇ ) is contained in a group G = SL (2 , C ) · Sp (2 n, C ) acting on 4 n -dimensional complex space C n = C ⊗ C n , with SL (2 , C ) acting tautologically on the first tensor factor, and Sp (2 n, C ) onthe second tensor factor.Trisymplectic structures occur naturally in hyperk¨ahler geometry, for the– 17 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties following reason. Let M be a hyperk¨ahler manifold, and Tw( M ) its twistorspace, that is, a total space of its twistor family (Definition 1.3). Twistorspace is a complex manifold, but it is non-algebraic and non-K¨ahler when M is compact.In [HKLR] (see also [S] and [V5]) the twistor space was used to define(possibly singular) hyperk¨ahler varieties. It turns out that a componentSec( M ) of the moduli of rational curves on Tw( M ) is equipped with a realstructure, in such a way that a connected component Sec R ( M ) of its setof real points is identified with M . Then one can describe the hyperk¨ahlerstructure on M as a certain geometric structure on Sec R ( M ).In [JV2] it was shown that Sec( M ) is equipped with a trisymplectic struc-ture, and its restriction to M = Sec R ( M ) gives the triple of symplectic forms ω I , ω J , ω K . One can understand this result by identifying Sec( M ) and a com-plexification of M . Then the trisymplectic structure on Sec( M ) is obtainedas a complexification of the triple ω I , ω J , ω K .A related notion of even Clifford structures on Riemannian manifoldswas introduced by A. Moroianu and U. Semmelmann in [MS]. An evenClifford structure on a Riemannian manifold M is a sub-bundle A ⊂ End(
T M ) which is closed under multiplication and fiberwise isomorphic toan even Clifford algebra acting on the tangent bundle
T M orthogonally. Aneven Clifford structure is called parallel if A is preserved by the Levi-Civitaconnection. If A is trivial, this Clifford structure is called flat . Moroianu andSemmelmann classified all manifolds admitting a parallel Clifford structure.If we are given a k -symplectic structure, then it induces an action of aneven Clifford algebra, as one can see from Proposition 2.11. However, themetric induced by symplectic forms and the Clifford algebra action is notnecessarily positive definite. If one removes the positive definiteness condi-tion, then any k -symplectic structure induces an even, flat Clifford structure,in the sense of Moroianu-Semmelmann.For k = 3, the corresponding flat Clifford structure is also preserved bythe Levi-Civita connection, hence parallel ([JV2]). It is not clear whetherthe same is true for k > version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties The notion of a k -symplectic manifold is certainly an intriguing one, and stillvery little understood.Let Ω be a k -symplectic structure on M . A general l -dimensional sub-space Ω ′ ⊂ Ω obviously gives an l -symplectic structure on M . This means,in particular, that any k -symplectic manifold is equipped with a 3( k − R of 3-symplectic structures. As shown in [JV2], a 3-symplectic manifold is equipped with a canonical torsion-free connectionpreserving the 3-symplectic structure (this connection is called the Chernconnection ; not to be confused with the Chern connection defined on holo-morphic Hermitian bundles). Given a 3-symplectic structure r ∈ R , denotethe corresponding Chern connection by ∇ r . Question 3.1:
Are all ∇ r equal for all r ∈ R ? If so, what is the holonomyof the Chern connection associated with a k -symplectic structure this way? Question 3.2:
Let M be a complex manifold equipped with a real struc-ture ι , and Ω an ι -invariant complex k -symplectic structure. The real part Ω R restricted to the real part M ι gives a k -dimensional space of symplecticforms of constant rank. Degenerate forms ω ∈ Ω correspond to the realpoints in the quadric Q . However, if the set of real points of Q is empty, allforms in Ω R ⊂ Λ R ( M ι ) are non-degenerate. When k = 3, Ω defines a hy-perk¨ahler structure on M ι , and any hyperk¨ahler structure can be defined thisway. What happens if we start with k -symplectic structure? Presumably,we obtain a generalization of hyperk¨ahler structures, with the quaternionalgebra replaced by a bigger-dimensional Clifford algebra. Such a manifoldwould admit a very special family of hyperk¨ahler structures, parametrizedby a Grassmanian of 3-dimensional subspaces in a k -dimensional space.In [JV2], a geometric reduction construction was established for trisym-plectic structures. Trisymplectic quotient was defined in such a way that atrisymplectic quotient of a trisymplectic manifold M equipped with an actionof d -dimensional compact Lie group G preserving the trisymplectic structureis a (dim M − d )-dimensional complex manifold equipped with a trisymplec-tic structure. This construction can be considered as a complexification ofthe hyperk¨ahler reduction of [HKLR]. Question 3.3:
Is there a geometric reduction construction for k -symplecticstructures? – 19 – version 3.2, Febr. 8, 2019 . Soldatenkov, M. Verbitsky k -symplectic structures and absolutely trianalytic subvarieties Acknowledgement:
We are grateful to K. Oguiso for interesting andfruitful discussions and to F. Bogomolov, D. Kaledin and A. Kuznetsov foruseful comments.
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Andrey SoldatenkovInstitut f¨ur MathematikHumboldt-Universit¨at zu BerlinUnter den Linden 610099 Berlin [email protected]
Misha VerbitskyInstituto Nacional de Matem´atica Pura e Aplicada (IMPA)Estrada Dona Castorina, 110Jardim Botˆanico, CEP 22460-320Rio de Janeiro, RJ - Brasilalso:Laboratory of Algebraic Geometry,National Research University HSE,Department of Mathematics, 7 Vavilova Str. Moscow, Russia, [email protected] . – 22 –– 22 –