Nonnegativity of signed Euler characteristics of moduli of curves and abelian varieties
Donu Arapura, Deepam Patel
TL;DR
The paper proves that the orbifold Euler characteristic of any perverse sheaf on the moduli stacks of curves ${\mathscr M}_{g,n}$ and principally polarized abelian varieties ${\mathscr A}_g$ is nonnegative in characteristic zero, extending Harer–Zagier and Harder-type formulas. The authors combine a nef-log geometric strategy for the abelian case—via log Dubson–Kashiwara, Fulton–Lazarsfeld positivity, and Beilinson-type arguments—with a Beilinson gluing and deformation-to-the-normal-cone approach for curves, together with an extension of the Torelli map to compact-type boundary strata to run an induction on boundary components. They also establish that the positivity results fail in positive characteristic, highlighting the essential role of char 0 and the existence of supersingular strata. The work yields a robust positivity framework for Euler characteristics of perverse sheaves on Shimura-type stacks and related moduli spaces, with potential broader applicability to other noncompact or stacky contexts. A key upshot is a unified, modular-geometry-based mechanism to deduce nonnegativity of Euler characteristics from nefness properties and controlled boundary behavior.
Abstract
Given a perverse sheaf on the moduli stack of principally polarized abelian varieties or the moduli stack of smooth curves with n marked points over a field of characteristic zero, we prove that the (orbifold) Euler characteristic is nonnegative. For constant coefficients, this follows immediately from formulas of Harer-Zagier and Harder. Our proof is different and in the case of abelian varieties uses log Dubson-Kashiwara plus the fact that Hodge bundles are nef. For curves, we require an additional inequality established using Beilinson's gluing construction. The first main result is shown to be false in positive characteristic.