Nonvarying, affine, and extremal geometry of strata of differentials

TL;DR

This paper demonstrates that certain strata of differentials are affine varieties with trivial tautological rings, and explores their geometric and dynamical properties, including extremality of stratification and divisor class relations.

Contribution

It establishes the affine nature and trivial tautological rings of nonvarying strata, and shows extremal properties of the stratification of the Hodge bundle.

Findings

Nonvarying strata have trivial tautological rings.

Strata of k-differentials of infinite area are affine varieties.

Merging zeros leads to extremal effective divisors.

Abstract

We prove that the nonvarying strata of abelian and quadratic differentials in [CM1, CM2] have trivial tautological rings and are affine varieties. We also prove that strata of $k$-differentials of infinite area are affine varieties for all $k$. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichm\"uller dynamics.