On the stabilization of the Betti numbers of the moduli space of sheaves on $\mathbb{P}^2$

TL;DR

This paper proves that the Betti numbers of certain moduli spaces of sheaves on the projective plane stabilize when the second Chern class exceeds a specific bound, advancing understanding of their topological properties.

Contribution

It establishes a stabilization result for the Betti numbers of moduli spaces of sheaves on , providing explicit bounds for when stabilization occurs.

Findings

Betti numbers stabilize for large second Chern class

Explicit bounds depend on rank and first Chern class

Results apply to moduli spaces on 

Abstract

Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$, then the $2n$th Betti number of the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.