Ergodic complex structures on hyperkahler manifolds

Abstract

Let $M$ be a compact complex manifold. The corresponding Teichmuller space $\Teich$ is a space of all complex structures on $M$ up to the action of the group of isotopies. The group $\Gamma$ of connected components of the diffeomorphism group (known as the mapping class group) acts on $\Teich$ in a natural way. An ergodic complex structure is the one with a $\Gamma$-orbit dense in $\Teich$. Let $M$ be a complex torus of complex dimension $\ge 2$ or a hyperkahler manifold with $b_2>3$. We prove that $M$ is ergodic, unless $M$ has maximal Picard rank (there is a countable number of such $M$). This is used to show that all hyperkahler manifolds are Kobayashi non-hyperbolic.

25 pages, last version (many minor corrections; one gap fixed)

References in corpus (3)

• Gromov-Hausdorff collapsing of Calabi-Yau manifolds
• Rational curves and special metrics on twistor spaces
• Families of Lagrangian fibrations on hyperkaehler manifolds

Cited by in corpus (3)

• Integral Transforms and Deformations of K3 Surfaces
• Kobayashi pseudometric on hyperkahler manifolds
• Teichmuller spaces, ergodic theory and global Torelli theorem