Rigid, not infinitesimally rigid surfaces with ample canonical bundle

Abstract: It was a long-standing problem of Morrow and Kodaira whether there are compact complex manifolds X with Def(X) a non reduced point. The first examples answering this question in the affirmative were given by Bauer and Pignatelli in 2018. As explained by Roberto Pignatelli in this seminar some weeks ago, these are certain surfaces of general type that have nodal canonical models. These canonical models are rigid AND infinitesimally rigid, while their desingularizations are still rigid, but not infinitesimally rigid anymore. One can therefore ask, if this situation is typical for rigid, not infinitesimally rigid surfaces of general type, or if it is possible to have examples with smooth canonical models. We answer this question also in the affirmative by constructing such a surface X via line arrangements and abelian covers. This construction was inspired by Vakil's version of „Murphy’s law in algebraic geometry“. (This is joint work with Christian Böhning and Roberto Pignatelli.)

algebraic geometry

Audience: researchers in the topic

Warwick algebraic geometry seminar