Virtual volumes of strata of meromorphic differentials with simple poles

TL;DR

This paper defines and computes volumes of strata of meromorphic differentials with simple poles, extending known holomorphic cases, and links these volumes to integrable systems through generating series.

Contribution

It introduces a new algebraic volume definition for meromorphic strata with simple poles and establishes methods for their computation and connection to integrable PDEs.

Findings

Volumes can be computed inductively on genus and singularities.

For strata with one zero, generating series solve a specific integrable PDE.

The volume definition aligns with Masur-Veech volumes in the holomorphic case.

Abstract

We work over strata of meromorphic differentials with poles of order 1, and on affine subspaces defined by linear conditions on the residues. We propose a definition of the volume of these objects as the integral of a tautological class on the projectivization of the stratum. By previous work with Chen-M\"oller-Zagier, this definition agrees with the Masur-Veech volumes in the holomorphic case. We show that these algebraic constants can be computed by induction on the genus and number of singularities. Besides, for strata with a single zero, we prove that the generating series of these volumes is a solution of an integrable system associated with the PDE: $u_tu_{xx}=u_tu_x+u_t - 1$.