k-symplectic structures and absolutely trianalytic subvarieties in hyperkahler manifolds
Let (M,I,J,K) be a hyperkahler 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 hyperk\"ahler 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 a normalization Z′ of a trianalytic subvariety is smooth; we prove that b2(Z′) is no smaller than b2(M) when M has maximal holonomy (that is, M is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the 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 hyperkahler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k=b2(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 hyperkahler manifold M with b2(M)≥2r+1 is at least 2r−1-dimensional.