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

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.