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Virtual volumes of strata of meromorphic differentials with simple poles

Adrien Sauvaget

TL;DR

The paper extends Masur–Veech volumes to strata of meromorphic differentials with simple poles by defining an algebraic volume via intersection theory on the projectivized stratum. It develops a full inductive framework in genus and in the number of zeros, expressing all volumes through refined polynomials a(μ) built from base data a((k)) and organized by boundary graphs and Segre classes. It provides explicit volume recursions for the numbers of zeros and establishes an integrable hierarchy for minimal strata, connecting residue-constraint degenerations to a PDE system that includes $u_tu_{xx}=u_tu_x+u_t-1$. The approach combines detailed cohomological machinery (back-bone graphs, Segre classes, boundary strata) with combinatorial recursive formulas, yielding both exact recursion relations and asymptotic insights in large genus; it also links to Siegel–Veech-type constants in special cases and opens avenues for higher-order-differentials generalizations.

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-Möller-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: $u_tu_{xx}=u_tu_x+u_t - 1$.

Virtual volumes of strata of meromorphic differentials with simple poles

TL;DR

The paper extends Masur–Veech volumes to strata of meromorphic differentials with simple poles by defining an algebraic volume via intersection theory on the projectivized stratum. It develops a full inductive framework in genus and in the number of zeros, expressing all volumes through refined polynomials a(μ) built from base data a((k)) and organized by boundary graphs and Segre classes. It provides explicit volume recursions for the numbers of zeros and establishes an integrable hierarchy for minimal strata, connecting residue-constraint degenerations to a PDE system that includes . The approach combines detailed cohomological machinery (back-bone graphs, Segre classes, boundary strata) with combinatorial recursive formulas, yielding both exact recursion relations and asymptotic insights in large genus; it also links to Siegel–Veech-type constants in special cases and opens avenues for higher-order-differentials generalizations.

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-Möller-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: .
Paper Structure (16 sections, 8 theorems, 72 equations)

This paper contains 16 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Strata of meromorphic differentials
  3. Masur--Veech volumes
  4. Volumes of strata as intersection numbers
  5. Combinatorics of virtual volumes
  6. An integrable hierarchy for strata with a single zero
  7. Acknowledgements
  8. Moduli spaces of differentials and their compactification
  9. Back-bone graphs
  10. Segre classes of strata
  11. Boundary components
  12. Description of $\overline{\mathcal{M}}_{-1}$
  13. Induction formulas in cohomology
  14. Induction on the genus for minimal strata
  15. Volume recursion for the numbers of zeros
  16. ...and 1 more sections

Key Result

Proposition 1.1

The value $a(\mu,\rho)_i$ is independent of $i$ and thus is denoted by $a(\mu,\rho)$.

Theorems & Definitions (25)

  • Proposition 1.1: see Section \ref{['ssec:cohomology']}
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 15 more