Seminar in Real & Complex Geometry
Monday, 15.07.2013, 12:0013:30,
Schreiber
building, room 210
Misha Verbitsky,
High School of Economics, Moscow
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
nonhyperbolic.
