Trianalytic subvarieties of generalized Kummer varieties

Abstract

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hyperkaehler structure. A finite cover of $T^{[n]}$ is a product of $T$ and a simply connected hyperkaehler manifold $K^{[n-1]}$, called generalized Kummer variety. We show that for $T$ generic, the corresponding generalized Kummer variety has no trianalytic subvarieties. This implies that a generic deformation of the generalized Kummer variety has no proper complex subvarieties.

22 pages, LaTeX 2e

Cited by in corpus (5)

• Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties
• Subvarieties of hypercomplex manifolds with holonomy in SL(n,H)
• k-symplectic structures and absolutely trianalytic subvarieties in hyperkahler manifolds
• Holomorphic symplectic geometry and orbifold singularities
• Local structure of hyperkaehler singularities in O'Grady's examples