Nonnegativity of signed Euler characteristics of moduli of curves and abelian varieties

TL;DR

This paper proves that the orbifold Euler characteristic of certain moduli stacks of curves and abelian varieties is nonnegative in characteristic zero, using novel methods involving log Dubson-Kashiwara and nef Hodge bundles.

Contribution

It introduces a new proof for the nonnegativity of Euler characteristics in characteristic zero, distinct from previous formulas, and highlights the failure of this result in positive characteristic.

Findings

Euler characteristic is nonnegative in characteristic zero

New proof uses log Dubson-Kashiwara and nef Hodge bundles

Result does not hold in positive characteristic

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.