Euler characteristics of moduli of twisted sheaves on Enriques surfaces

TL;DR

This paper computes the virtual Euler characteristic of moduli spaces of stable twisted sheaves on Enriques surfaces, revealing a dependence solely on the virtual dimension and relating it to Hilbert schemes of points.

Contribution

It establishes a precise formula for the virtual Euler characteristic of these moduli spaces, linking it to the geometry of Enriques surfaces and Hilbert schemes.

Findings

e^{vir}(M) = 0 for odd N

e^{vir}(M) = 2 * e(Y^{[N/2]}) for even N

Virtual Euler characteristic depends only on the virtual dimension

Abstract

Let $Y$ be an Enriques surface and let $\mathcal{A}$ be an Azumaya algebra corresponding to the non-trivial Brauer class. Let $M$ be the moduli space of stable twisted sheaves on Enriques surfaces with twisted Chern character $M^H_{\mathcal{A}/Y}(2,c_1,\operatorname{ch}_2)$ with virtual dimension $N$. We show that the virtual Euler characteristic $e^\mathrm{vir}(M)$ only depends on $N$, more precisely, $e^\mathrm{vir}(M)=0$ when $N$ is odd and $e^\mathrm{vir}(M)=2\cdot e(Y^{[\frac{N}{2}]})$ when $N$ is even.