k-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Abstract

a b s t r a c t Let (M, I, J, K ) be a hyperkähler manifold, and Z ⊂ (M, I) a complex subvariety in (M, I).We say that Z is trianalytic if it is complex analytic with respect to J and K , and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures (M, I, J ′ , K ′ ) containing I. For a generic complex structure I on M, all complex subvarieties of (M, I) are absolutely trianalytic. It is known that the normalization Z ′ of a trianalytic subvariety is smooth; we prove thatTo study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d 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 hyperkähler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k = b 2 (M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k − 1). Then an absolutely trianalytic torus in a hyperkähler manifold M with b 2 (M) 2r + 1 is at least 2 r−1 -dimensional.

Kuga-Satake construction and cohomology of hyperkähler manifolds

“…Consider the polynomial p ∈ S 2n W * given by p(ω) = ω ∧2n (here we identify Λ 4n V * with k). Analogously to [SV2,Lemma 3.10] we can prove that W is a k-symplectic structure.…”

“…The k-symplectic spaces were introduced in [SV2]; however the corresponding geometric structure for k = 3 was known since 1990-ies under the name "hypersymplectic". A hypersymplectic space ( [DS]) is a vector space V over a field k = R or C equipped with a triple of symplectic forms ω 1 , ω 2 , ω 3 in such a way that the operators ω i • ω −1 j : V −→ V generate the matrix algebra Mat(2, k).…”

“…Hypersymplectic structures were generalized to k-symplectic structures in [SV2]. The starting point was again hyperkähler geometry, but an entirely different facet.…”

Transcendental Hodge algebra

“…The formalism of transcendental Hodge algebra seems to be particularly suited for the study of the holomorphically symplectic subvarieties. In this paper, we generalize the results of [SV2] from trianalytic to more general symplectic subvarieties. In [SV2] it was shown that a trianalytic complex subtorus Z in a very general deformation of a hyperkähler manifold M satisfies dim Z 2 ⌊ d+1 2 ⌋ , where d = b 2 − 2 is the dimension of the universal family of deformations of M .…”

“…In this paper, we generalize the results of [SV2] from trianalytic to more general symplectic subvarieties. In [SV2] it was shown that a trianalytic complex subtorus Z in a very general deformation of a hyperkähler manifold M satisfies dim Z 2 ⌊ d+1 2 ⌋ , where d = b 2 − 2 is the dimension of the universal family of deformations of M . In this paper, the same result is proven for projective M generic in a d-dimensional family of deformations (Corollary 7.6).…”