Asymptotic Behaviors of Moduli of One-dimensional Sheaves on Surfaces

Abstract

In this paper, we study the asymptotic behaviors of the Betti numbers and Picard numbers of the moduli space $M_{\beta ,\chi }$ of one-dimensional sheaves supported in a curve class $\beta$ on $S$ with Euler characteristic $\chi$. We determine the intersection cohomology Betti numbers of $M_{\beta ,\chi }$ when $S$ is a del Pezzo surface and $\beta$ is sufficiently positive. As an application, we formulate a $P=C$ conjecture regarding the refined BPS invariants for local del Pezzo surfaces.

28 pages, comments are very welcome!