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Non-trivial and Non-tautological Cohomology of Strata of Differentials

Dawei Chen, Prabhat Devkota, Samuel Grushevsky, Martin Möller

Abstract

In this paper we construct various non-trivial and non-tautological cohomology classes on compactified and uncompactified strata of curves with a differential, by using the geometry of the boundary stratification of the moduli space of multi-scale differentials.

Non-trivial and Non-tautological Cohomology of Strata of Differentials

Abstract

In this paper we construct various non-trivial and non-tautological cohomology classes on compactified and uncompactified strata of curves with a differential, by using the geometry of the boundary stratification of the moduli space of multi-scale differentials.
Paper Structure (7 sections, 17 theorems, 62 equations, 7 figures)

This paper contains 7 sections, 17 theorems, 62 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Non-trivial classes in $H^3({\overline\cH})$
  4. Non-tautological even degree cohomology classes.
  5. Meromorphic strata with $H^2(\cH_{g,n}(\mu))\ne 0$
  6. Proof of \ref{['prop:genusLT']}
  7. Linear independence and extremality of irreducible components of boundary divisors

Key Result

Theorem 1.1

For any stratum in genus $g\ge 3$ (and for $g=2$ provided $n\ge 3$) such that $m_1\ge 9$, the third cohomology is non-zero: $H^3(\overline\cH_{g,n}(\mu))\ne 0$. Moreover, for $n$ and $m_2,\dots,m_n$ fixed, $\dim H^3(\overline\cH_{g,n}(\mu))$ grows at least exponentially in $m_1$ (or equivalently, in

Figures (7)

  • Figure 1: Level graph with a genus $1$ residueless stratum in the bottom level
  • Figure 2: Divisors containing $D_\Lambda$
  • Figure :
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition A.1
  • Proposition A.2
  • Proposition 2.1
  • ...and 35 more