The Betti numbers of the moduli space of stable sheaves of rank 3 on P2

TL;DR

The paper advances the understanding of moduli spaces of stable sheaves on rational surfaces by computing generating functions for the Betti numbers of rank $3$ sheaves on $\mathbb{P}^2$ and its blow-up. It combines wall-crossing (including semi-primitive cases), the blow-up formula, and flow-tree insights to derive compact expressions in terms of modular forms and indefinite theta functions on lattices of signature $(2,2)$. The results extend the established rank $2$ program to rank $3$, supply explicit Betti-number data for low $c_2$, and reinforce the connection between geometric invariants and physical BPS-counts via modularity and attractor-flow intuition. This work deepens the link between enumerative geometry on rational surfaces and the modular structure expected from $\mathcal{N}=4$ gauge theory and Calabi-Yau compactifications, with potential extensions to higher ranks.

Abstract

This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on the projective plane P2 and its blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up. These can be derived equivalently using flow trees, which appear in the physics of BPS-states. The Betti numbers for P2 follow from those for the blow-up by the blow-up formula. The generating functions are expressed in terms of modular functions and indefinite theta functions.

Figures (1)

• Figure 1: The ample cone of $\mathbb{\tilde{P}}^2$, together with the three walls for $\Gamma=(2,-C-f,2)$, namely for $(a,b)=(1,0)$, $(2,0)$, $(3,0)$.

Theorems & Definitions (4)

• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.3