Stable homology of strata of abelian differentials

TL;DR

This paper proves that the homology of strata of abelian differentials stabilizes when the number of simple zeros is large, and shows that their rational cohomology aligns with tautological classes, with trivial rational Picard group in certain cases.

Contribution

It introduces an $h$-principle for these strata, establishing homological stability and cohomological properties, extending to higher order differentials.

Findings

Homology stabilizes when simple zeros are numerous.

Rational cohomology matches tautological classes in the stable range.

Rational Picard group is trivial for unprojectivized strata.

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.