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Independence of tautological classes and cohomological stability for strata of differentials

Dawei Chen, Hannah Larson

Abstract

The tautological rings of strata of differentials are known to be generated by divisor classes. In this paper, we give lower bounds on the degrees of relations among them, depending on the genus $g$ and the number of simple zeros. For strata with more than $4g/3$ simple zeros, our results show that there are no relations in degrees less than $\lfloor g/3 \rfloor + 1$. Moreover, we conjecture that, outside of a few exceptions, there is always a non-trivial relation in degree $\lfloor g/3 \rfloor + 1$, and prove the conjecture for all strata of holomorphic abelian differentials with $g \leq 30$. We also prove that the cohomology rings of strata of holomorphic differentials with sufficiently many simple zeros stabilize to the free algebra on the tautological divisor class. Finally, we show that for a large class of holomorphic abelian strata, containing hyperelliptic differentials, the tautological ring is non-trivial for sufficiently large $g$.

Independence of tautological classes and cohomological stability for strata of differentials

Abstract

The tautological rings of strata of differentials are known to be generated by divisor classes. In this paper, we give lower bounds on the degrees of relations among them, depending on the genus and the number of simple zeros. For strata with more than simple zeros, our results show that there are no relations in degrees less than . Moreover, we conjecture that, outside of a few exceptions, there is always a non-trivial relation in degree , and prove the conjecture for all strata of holomorphic abelian differentials with . We also prove that the cohomology rings of strata of holomorphic differentials with sufficiently many simple zeros stabilize to the free algebra on the tautological divisor class. Finally, we show that for a large class of holomorphic abelian strata, containing hyperelliptic differentials, the tautological ring is non-trivial for sufficiently large .
Paper Structure (11 sections, 14 theorems, 105 equations, 1 figure)

This paper contains 11 sections, 14 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The partial compactification $\overline{X}^\ell(m_1, \ldots, m_k)$
  3. Notation and Borel--Moore homology
  4. Relating $\mathcal{P}^\ell(\mu)$ to a projective bundle
  5. The case $\ell = 1$
  6. The case $\ell \geq 2$
  7. The stable cohomology of $\overline{X}^{\ell}(m_1, \ldots, m_k)$
  8. The pure weight cohomology of $X^\ell(m_1, \ldots, m_k)$
  9. The stable cohomology of $X^{\ell}(m_1, \ldots, m_k)$
  10. Upper bounds on the vanishing degree of $\eta$
  11. Hyperelliptic differentials and varying strata

Key Result

Theorem 1.1

Let $\mu = (m_1, \ldots, m_k, 1^{\ell(2g - 2) - m})$ with $m = m_1 + \cdots + m_k$ the sum of $k$ specified orders such that the remaining orders are all equal to $1$. Let $r$ be such that $m_i < 0$ for $i \leq r$ and $m_i \geq 0$ for $i > r$. If $m_i \neq -\ell$ for all $i$, then the surjection is an isomorphism in degrees $* \leq \min\{g/3, \ell(2g - 2) - m - g + \delta_{0r}\delta_{1\ell} - 1\}

Figures (1)

  • Figure 1: The top row and top three terms in the right column are exact. The blue arrows are both surjective when $* - 2 \leq \mathsf{s}^\ell_r(m+2)$ by Lemma \ref{['lem:stab']} and \ref{['open']}. Consequently, the kernel of $W_*H^*(\overline{X}) \to W_*H^*(X)$ is generated by the images of the two red maps to it.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 19 more