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Stable homology of strata of abelian differentials

Philip Tosteson

Abstract

We show that the homology of strata of abelian differentials stabilizes in a range where the number of simple zeros is large relative to the homological degree. In this range, we show that the rational cohomology agrees with the restriction of the tautological classes to the stratum, and that the rational Picard group is trivial for unprojectivized strata. Our proof method is to develop an $h$-principle for these strata, valid in a range of homological degrees that increases with the number of simple zeros. The same approach also applies to higher order differentials.

Stable homology of strata of abelian differentials

Abstract

We show that the homology of strata of abelian differentials stabilizes in a range where the number of simple zeros is large relative to the homological degree. In this range, we show that the rational cohomology agrees with the restriction of the tautological classes to the stratum, and that the rational Picard group is trivial for unprojectivized strata. Our proof method is to develop an -principle for these strata, valid in a range of homological degrees that increases with the number of simple zeros. The same approach also applies to higher order differentials.

Paper Structure

This paper contains 22 sections, 25 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Proof method and structure of paper
  3. Relation to other work
  4. Further Questions
  5. Acknowledgements
  6. Notation
  7. Posets of divisors of multiplicity $\geq 2$
  8. Definitions and combinatorics
  9. Spectral sequence associated to a continuous stratification
  10. Convergence criterion
  11. Dimension bounds
  12. Brill Noether bounds
  13. Proof of the $h$-principle
  14. Resolving algebraic sections
  15. Finite dimensional approximations
  16. ...and 7 more sections

Key Result

Theorem 1.1

For $0< i \leq m_1(\mu)/2 - 2g/3$, the rational cohomology $H^i(\mathcal{H}_{g,\mu}, \mathbb{Q})$ vanishes.

Theorems & Definitions (66)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 56 more