# Cohomology classes of strata of differentials

## TL;DR

This paper develops a framework for studying the cohomology classes of strata of meromorphic differentials with prescribed poles, providing algorithms for their computation and insights into their Picard groups.

## Contribution

It introduces a tautological cohomology ring for spaces of meromorphic differentials and provides an algorithm to compute the cohomology classes of all strata.

## Key findings

- All strata classes are tautological.
- An explicit algorithm for class computation is provided.
- Relations in the Picard groups of strata are deduced.

## Abstract

We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar\'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.