Euler characteristics of strata of k-differentials
Abstract
We compute the orbifold Euler characteristics of strata of k-differentials on the moduli space of smooth pointed curves. Our main result is a closed coefficient-extraction formula in terms of hyperbolic functions, valid for arbitrary strata and generalizing the Harer--Zagier formula for the Euler characteristic of the moduli space of curves. In particular, these Euler characteristics are polynomial in the prescribed orders at the marked points. The proof first uses Riemann--Roch and Serre duality to reduce the computation to the Hurwitz case k=0, which is then evaluated using the representation theory of the symmetric group and Fock-space techniques. We also give an intersection-theoretic expression for the same invariants as tautological integrals over double ramification cycles and derive a recursion which uniquely determines them from genus zero. For odd k and odd orders, we introduce a spin-weighted refinement and formulate corresponding closed and intersection-theoretic conjectural expressions. Finally, we determine the large-genus asymptotics of the Euler characteristics.
Related
- Asymptotics of Masur-Veech volume of strata of quadratic differentials with many simple poles
- Cohomology ring of the strata of holomorphic differentials of genus 3
- Semipositivity and complete subvarieties of projectivised strata of holomorphic differentials
- Perverse Euler Characteristics of Hermitian Locally Symmetric Spaces
- Moduli Spaces of Twisted Endomorphisms and Euler Characteristics
BibTeX
@misc{costantini2026euler,
title = {{Euler characteristics of strata of $k$-differentials}},
author = {Matteo Costantini and Alessandro Giacchetto and Martin Möller and Adrien Sauvaget and Johannes Schmitt},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26922},
eprint = {2609.26922},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05840686v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler characteristics of strata of k-differentials
AU - Matteo Costantini
AU - Alessandro Giacchetto
AU - Martin Möller
AU - Adrien Sauvaget
AU - Johannes Schmitt
PY - 2026
DA - 2026/09/22
PB - HAL
DO - 10.48550/arxiv.2609.26922
UR - https://hal.science/hal-05840686v1
AB - We compute the orbifold Euler characteristics of strata of k-differentials on the moduli space of smooth pointed curves. Our main result is a closed coefficient-extraction formula in terms of hyperbolic functions, valid for arbitrary strata and generalizing the Harer--Zagier formula for the Euler characteristic of the moduli space of curves. In particular, these Euler characteristics are polynomial in the prescribed orders at the marked points. The proof first uses Riemann--Roch and Serre duality to reduce the computation to the Hurwitz case k=0, which is then evaluated using the representation theory of the symmetric group and Fock-space techniques. We also give an intersection-theoretic expression for the same invariants as tautological integrals over double ramification cycles and derive a recursion which uniquely determines them from genus zero. For odd k and odd orders, we introduce a spin-weighted refinement and formulate corresponding closed and intersection-theoretic conjectural expressions. Finally, we determine the large-genus asymptotics of the Euler characteristics.
ER -