Speaker
Michael Temkin
Abstract
The projectivized Hodge bundle on M_g can be naturally stratified by the pattern of zeros and poles, and then a natural question is to find a nice, e.g. smooth and modular, compactification of these strata over the nodal boundary of M_g. Bainbridge, Chen, Gendron, Grushevsky and Möller studied this question in a series of papers by using complex-analytic techniques: first they described an incidence compactification, obtained by taking the schematic closure in the projectivized Hodge bundle of barM_{g,n}, then they extended these results to k-differentials, and in the last work they refined the incidence compactification to a smooth compactification with a modular interpretation.
A different proof of the characterization of incidence compactification was found in my work with I. Tyomkin and it was very recently extended by U. Brezner to k-differentials. These proofs are based on Berkovich geometry, but they have many common features with [BCGGM1] and [BCGGM2], first of all due to patching via certain good coordinates, though they provide a new interpretation of the global residue condition. In this talk I will briefly outline this story and then proceed to a work in progress with Tyomkin, where we construct a modular compactification by tools of log geometry. Nothing is written down yet, but as it seems now one will obtain a simple construction of a log modular compactification, which applies in any characteristic. It is (combinatorially) coarser than the construction of [BCGGM3] and is hopefully log smooth, but certainly not smooth. Also, it makes no use of good coordinates and residue conditions.