The moduli space of multi-scale differentials

2019/10/29 by Matt Bainbridge, Dawei Chen, Bainbridge, Matt +7 · 26 citations
Mathematics · #Abelian group #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Compactification (mathematics) #Computer science #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Moduli #Moduli of algebraic curves #Moduli space #Orbifold #Physics #Pure mathematics #Stack (abstract data type)

paper · pdf · doi:10.48550/arxiv.1910.13492

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2019/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined in [BCGGM18] as the closure of the stratum of abelian differentials in the closure of the Hodge bundle. We also define families of projectivized multi-scale differentials, which gives a proper Deligne-Mumford stack, and our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the GL₂(\mathbb R)-action in the interior of the moduli space extends continuously to the boundary.

Cited by

• Realizability of tropical pluri-canonical divisors
• Isoresidual fibrations and the birational geometry of moduli of pointed stable curves
• Finite isoresidual covers in strata of k-differentials
• Categorical realization of collapsing subsurfaces and perverse schobers
• Reconstructing orbit closures from their boundaries
• Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
• Pixton's formula and Abel-Jacobi theory on the Picard stack
• Counting geodesics on prime-order k-differentials
• The Incidence Variety Compactification of strata of d-differentials in genus 0
• Ends of the strata of differentials
• The \mathbbS_n-equivariant Chow polynomial of the braid matroid
• Reduction and lifting problem for differential forms on Berkovich curves
• Multiscale differentials and wonderful models
• Area Siegel--Veech constants for affine invariant submanifolds of REL zero
• Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

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• The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials
• Principal boundary of moduli spaces of abelian and quadratic differentials