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 $g$ 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 $g$.