Moduli spaces of varieties of general type are naturally of log general type

Abstract

We generalize work of Popa-Schnell on Viehweg hyperbolicity for smooth projective varieties to the case of smooth Deligne-Mumford stacks. The results of Popa-Schnell build on results of Viehweg-Zuo, utilize a result of Campana-Păun, and extend results of Kebekus-Kovács and Patakfalvi. We show that if a smooth proper DM stack with projective coarse moduli space parameterizes a family of varieties of general type having maximal variation, then the natural log canonical bundle of the stack is big. We also consider implications for the coarse moduli space of the stack, and apply the results to several standard moduli spaces.