Spin and hyperelliptic structures of log twisted differentials

TL;DR

This paper introduces a new framework for compactifying and analyzing the boundary structures of moduli spaces of abelian differentials using log geometry, revealing spin and hyperelliptic features.

Contribution

It extends abelian differentials to the boundary via log twisted differentials, providing a compactification that captures spin and hyperelliptic structures.

Findings

Spin parity can be distinguished on the boundary

Log twisted hyperelliptic differentials form a toroidal compactification

The moduli stack captures boundary behaviors of abelian differentials

Abstract

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected components, due to spin and hyperelliptic structures. We prove that the spin parity can be distinguished on the boundary of the log compactification. Moreover, combining the techniques of log geometry and admissible covers, we introduce log twisted hyperelliptic differentials, and prove that their moduli stack provides a toroidal compactification of the hyperelliptic loci in the open strata.