Ergodic complex structures on hyperkahler manifolds: an erratum

Abstract

Let $M$ be a hyperkahler manifold, $\Gamma$ its mapping class group, and $Teich$ the Teichmuller space of complex structures of hyperkahler type. After we glue together birationally equivalent points, we obtain the so-called birational Teichmuller space $Teich_b$. Every connected component of $Teich_b$ is identified with its period space $P$ by global Torelli theorem. The mapping class group of $M$ acts on $P$ as a finite index subgroup of the group of isometries of the integer cohomology lattice, that is, satisfies assumptions of Ratner theorem. We prove that there are three classes of orbits, closed, dense and the intermediate class which corresponds to varieties with $Re(H^{2,0}(M))$ containing a given rational vector. The closure of the later orbits is a fixed point set of an anticomplex involution of $P$. This fixes an error in the paper 1306.1498, where this third class of orbits was overlooked. We explain how this affects the works based on 1306.1498.