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Chern classes of linear submanifolds with application to spaces of k-differentials and ball quotients

Matteo Costantini, Martin Möller, Johannes Schwab

TL;DR

The work provides a unified framework to compute the Chern classes of the cotangent bundle for linear submanifolds of moduli spaces of Abelian differentials, extending to strata of $k$-differentials via multi-scale compactifications. It develops a detailed boundary-graph calculus, including level graphs, prong-matchings, and push-pull formulas, to express topological invariants like the Euler characteristic in terms of tautological data. The authors implement these intersection-theory ingredients in a Sage-based package and apply them to key problems: (i) Euler characteristics of eigenform loci and Teichmüller curves, (ii) uniform formulas for strata of $k$-differentials, and (iii) an algebraic proof that certain compactifications are ball quotients under the INT condition, notably in genus zero with five marked points. The results connect with Pixton-type formulas for fundamental classes, provide cross-checks against Masur–Veech volume data, and yield a computational toolkit (diffstrata) to explore tautological evaluations, deepening the link between geometry of moduli spaces and ball-quotient theory. The work thus offers both conceptual advances in compactifications of linear submanifolds and practical computational tools for intersection theory in moduli spaces of differentials.

Abstract

We provide formulas for the Chern classes of linear submanifolds of the moduli spaces of Abelian differentials and hence for their Euler characteristic. This includes as special case the moduli spaces of k-differentials, for which we set up the full intersection theory package and implement it in the sage-program diffstrata. As an application, we give an algebraic proof of the theorems of Deligne-Mostow and Thurston that suitable compactifications of moduli spaces of k-differentials on the 5-punctured projective line with weights satisfying the INT-condition are quotients of the complex two-ball.

Chern classes of linear submanifolds with application to spaces of k-differentials and ball quotients

TL;DR

The work provides a unified framework to compute the Chern classes of the cotangent bundle for linear submanifolds of moduli spaces of Abelian differentials, extending to strata of -differentials via multi-scale compactifications. It develops a detailed boundary-graph calculus, including level graphs, prong-matchings, and push-pull formulas, to express topological invariants like the Euler characteristic in terms of tautological data. The authors implement these intersection-theory ingredients in a Sage-based package and apply them to key problems: (i) Euler characteristics of eigenform loci and Teichmüller curves, (ii) uniform formulas for strata of -differentials, and (iii) an algebraic proof that certain compactifications are ball quotients under the INT condition, notably in genus zero with five marked points. The results connect with Pixton-type formulas for fundamental classes, provide cross-checks against Masur–Veech volume data, and yield a computational toolkit (diffstrata) to explore tautological evaluations, deepening the link between geometry of moduli spaces and ball-quotient theory. The work thus offers both conceptual advances in compactifications of linear submanifolds and practical computational tools for intersection theory in moduli spaces of differentials.

Abstract

We provide formulas for the Chern classes of linear submanifolds of the moduli spaces of Abelian differentials and hence for their Euler characteristic. This includes as special case the moduli spaces of k-differentials, for which we set up the full intersection theory package and implement it in the sage-program diffstrata. As an application, we give an algebraic proof of the theorems of Deligne-Mostow and Thurston that suitable compactifications of moduli spaces of k-differentials on the 5-punctured projective line with weights satisfying the INT-condition are quotients of the complex two-ball.
Paper Structure (29 sections, 45 theorems, 151 equations, 6 figures, 3 tables)

This paper contains 29 sections, 45 theorems, 151 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Applications: Teichmüller curves in genus two
  3. Strata of $k$-differentials
  4. Ball quotients
  5. Logarithmic differential forms and toric varieties
  6. The closure of linear submanifolds
  7. Linear submanifolds in generalized strata
  8. Multi-scale differentials: boundary combinatorics
  9. Multi-scale differentials: Prong-matchings and stack structure
  10. Decomposition of the logarithmic tangent bundle
  11. The closure of linear submanifolds
  12. Push-pull comparison for linear submanifolds
  13. Evaluation of tautological classes
  14. Vertical tautological ring
  15. Evaluation of $\xi_{\mathcal{H}}$
  16. ...and 14 more sections

Key Result

Theorem 1

The first Chern class of the logarithmic cotangent bundle of a projectivized compactified linear submanifold $\overline \mathcal{H}$ is where $N := \dim(\Omega \mathcal{H})$ and where $N_{\Gamma^+}^\top:=\dim(D^{\mathcal{H},\top}_{\Gamma^+})+1$ is the dimension of the unprojectivized top level stratum in $D^\mathcal{H}_{\Gamma^+}$.

Figures (6)

  • Figure 1: Euler characteristics of some minimal strata of primitive $k$-differentials. Note that in the case of $k=2$ the stratum of primitive minimal $k$-differentials is empty, see MaSmillie.
  • Figure 1: The boundary divisors of the eigenform locus $E$.
  • Figure 2: A covering of graphs $\pi : \widehat{\Gamma} \to \Gamma$ in $\Xi^{2}\overline{\mathcal{M}}^{}_{3,1}(8)$ with non-trivial $S(\pi)$.
  • Figure 2: Euler characteristics of the strata of primitive quadratic differentials in genus $2$ with at most one simple pole.
  • Figure 3: Euler characteristics of the strata of primitive holomorphic $3$-differentials in genus $2$.
  • ...and 1 more figures

Theorems & Definitions (85)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Bainbridge
  • Corollary 1
  • Proposition 1
  • Theorem 5
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 75 more