k-symplectic structures and absolutely trianalytic subvarieties in hyperkahler manifolds

Abstract

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ 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 $b_2(Z')$ is no smaller than $b_2(M)$ 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 $k=b_2(M)$. 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 hyperkahler manifold $M$ with $b_2(M)\geq 2r+1$ is at least $2^{r-1}$-dimensional.