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

TL;DR

The survey develops an algebro-geometric program for the topology of moduli spaces of differentials, focusing on strata as structured spaces with period coordinates, degenerations, and compactifications. It synthesizes key results on connected components (hyperelliptic and spin distinctions) and the K$(\pi,1)$-type questions for fundamental groups, alongside monodromy, deformation theory of singularities, and stability conditions. It then integrates these with Euler characteristic computations via isoperimetric and intersection-theoretic methods, and analyzes tautological rings and affine geometry to illuminate boundary structures and dimensions. Central contributions include a synthesis of known classifications across holomorphic, meromorphic, quadratic, and generalized strata; identification of major open problems (notably algebraic proofs of component classifications and $K(\pi,1)$ results beyond low genus); and a framework linking differential geometry, stability conditions, and boundary behavior to topological invariants. The work aims to guide future algebraic-geometric approaches to extend results to other fields, extend multi-scale compactifications, and deepen connections between topology, dynamics, and enumerative geometry in moduli spaces of differentials.

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.