kk-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Let (M,I,J,K)(M,I,J,K) be a hyperkähler manifold, and Z⊂(M,I)Z⊂(M,I) a complex subvariety in (M,I)(M,I). We say that ZZ is trianalytic if it is complex analytic with respect to JJ and KK, and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures (M,I,J′,K′)(M,I,J′,K′) containing II. For a generic complex structure II on MM, all complex subvarieties of (M,I)(M,I) are absolutely trianalytic. It is known that the normalization Z′Z′ of a trianalytic subvariety is smooth; we prove that b2(Z′)⩾b2(M)b2(Z′)⩾b2(M), when MM has maximal holonomy (that is, MM is IHS).To study absolutely trianalytic subvarieties further, we define a new geometric structure, called kk-symplectic structure; this structure is a generalization of hypersymplectic structure. A kk-symplectic structure on a 2d2d-dimensional manifold XX is a kk-dimensional space RR of closed 2-forms on XX which all have rank 2d2d or dd. It is called non-degenerate if the set of all degenerate forms in RR is a smooth, non-degenerate quadric hypersurface in RR.We consider absolutely trianalytic tori in a hyperkähler manifold MM of maximal holonomy. We prove that any such torus is equipped with a non-degenerate kk-symplectic structure, where k=b2(M)k=b2(M). We show that the tangent bundle TXTX of a kk-symplectic manifold is a Clifford module over a Clifford algebra Cl(k−1)Cl(k−1). Then an absolutely trianalytic torus in a hyperkähler manifold MM with b2(M)⩾2r+1b2(M)⩾2r+1 is at least 2r−12r−1-dimensional.