# Seminar in Real & Complex Geometry

## Ergodic complex structures

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 \$\geq 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.