Invariants of moduli spaces of stable sheaves on ruled surfaces

TL;DR

This paper calculates the Betti numbers of moduli spaces of stable sheaves on ruled surfaces, extending known formulas and confirming a conjecture for arbitrary rank sheaves.

Contribution

It generalizes existing formulas for Betti numbers of stable sheaves to arbitrary rank on ruled surfaces and verifies Manschot's conjecture for Hirzebruch surfaces.

Findings

Betti numbers computed for arbitrary rank sheaves

Generalization of Goettsche and Yoshioka formulas

Confirmation of Manschot's conjecture on Hirzebruch surfaces

Abstract

We compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the conjecture of Manschot for arbitrary rank sheaves on the Hirzebruch surfaces.