An algebro-geometric perspective on the topology of moduli spaces of differentials

TL;DR

This survey explores the algebro-geometric understanding of the topology of moduli spaces of differentials on Riemann surfaces, highlighting recent progress, open problems, and future directions in the field.

Contribution

It provides a comprehensive overview of known results and open questions on the topology of these moduli spaces from an algebro-geometric perspective.

Findings

Progress in computing invariants of moduli spaces

Classification of linear subvarieties

Understanding degenerations and compactifications

Abstract

Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials to appear in various guises across many areas, including algebraic geometry, dynamical systems, combinatorial enumeration, and mathematical physics. Over the past few decades, remarkable progress has been made in computing invariants of these moduli spaces, classifying linear subvarieties, understanding degenerations and compactifications, and developing intersection theory on these spaces. Despite these advances, our understanding of the topology of moduli spaces of differentials remains limited, and many fundamental questions are still open. In this survey, we aim to present, from an algebro-geometric perspective, the known results and open problems concerning the topology of moduli spaces of differentials, as well as their connections to other aspects of the field, with the hope of inspiring further developments in the coming decade.