The moduli space of curves — the space parameterizing all Riemann surfaces of a given genus — is one of the most studied objects in mathematics. But curves carry additional structure: abelian differentials, which are holomorphic one-forms that define flat metrics with conical singularities. The space of all (curve, differential) pairs stratifies according to the pattern of zeros of the differential. Each stratum is a complex orbifold with its own topology.
Tosteson (arXiv: 2603.24563) proves that the homology of these strata stabilizes. As the number of simple zeros grows (with the sum of zero orders fixed), the homology groups become independent of the specific zero pattern, up to a degree that increases linearly with the number of simple zeros.
In the stable range, the rational cohomology matches the tautological classes — the algebraically defined classes that come from the universal curve over the moduli space. Nothing exotic appears. And the rational Picard group (classifying line bundles) vanishes for unprojectivized strata. The topology simplifies dramatically once enough simple zeros are present.
The proof uses an h-principle — a technique from differential topology showing that a topological condition (the existence of a homotopy) is sufficient for a geometric condition (the existence of a differential form with prescribed zeros). The h-principle provides a simplicial model for the stratum that makes the stability visible.
The through-claim: complexity dilutes with multiplicity. Adding more simple zeros to an abelian differential makes the stratum's topology simpler, not more complex. The additional zeros provide enough flexibility that the topological constraints relax. The most complicated strata — the ones with few, high-order zeros — are the ones whose topology is genuinely rich. Fragmentation equals simplification.
Tosteson, 2603.24563. Algebraic geometry / moduli spaces / abelian differentials / homological stability / h-principle.