The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials

Abstract

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

An intersection-theoretic proof of the Harer–Zagier fomula

“…Firstly, let us recall a generalised Gauss-Bonnet formula expressing the orbifold Euler characteristic of certain open orbifolds as integrals of the Chern class of the logarithmic cotangent bundle. A proof of the formula can be found in [CMZ22].…”

Pixton’s formula and Abel–Jacobi theory on the Picard stack

“…Let X be a non-singular projective variety with line bundle L and an effective curve class β ∈H 2 (X, Z). Let ( 41 ) In the ongoing project [22], the authors study formulas for Euler characteristics of strata of differentials in terms of intersection numbers on the compactification of these strata constructed in [10].…”

Nonvarying, affine and extremal geometry of strata of differentials

“…In general little is known about the homology of strata of differentials (nevertheless see [ CMZ ] for computing the Euler characteristics and [ Zy ] for the -isodelaunay decomposition). Even if one is interested in strata of holomorphic differentials in high genus, differentials in low genus and meromorphic differentials naturally appear in the boundary of compactified strata ([ BCGGM1, BCGGM3 ]).…”