Bounds on Wahl singularities from symplectic topology

Abstract

A complex surface is said to have general type if its canonical bundle is big. The moduli space of surfaces of general type with fixed characteristic numbers $K^2$ and $\chi$ admits a compactification, constructed by Kolla ́r and Shepherd-Barron, whose boundary points correspond to surfaces with semi-log-canonical (slc) singularities, in much the way that the boundary points of Deligne-Mumford space correspond to nodal curves.