Cohomological stabilization, perverse filtrations, and refined BPS invariants for del Pezzo surfaces

TL;DR

This paper establishes an asymptotic product formula for refined BPS invariants of local del Pezzo surfaces, revealing cohomological stabilization of perverse filtrations and confirming conjectures in the case of the projective plane.

Contribution

It introduces a new asymptotic product formula for refined BPS invariants and demonstrates the stabilization of perverse filtrations, connecting them with Chern filtrations via Fourier transform.

Findings

Proves an asymptotic product formula for refined BPS invariants.

Shows cohomological stabilization of the perverse filtration.

Resolves conjectures for the projective plane case.

Abstract

We prove an asymptotic product formula for the refined BPS invariants associated with a local del Pezzo surface. Our formula governs the cohomological stabilization of the perverse filtration on the intersection cohomology of the moduli space of 1-dimensional semistable sheaves on a del Pezzo surface. Combined with the theory of Fourier transform of Maulik--Shen--Yin, we show that the perverse filtration matches asymptotically with the Chern filtration defined via tautological classes. In the case of the projective plane, our results resolve conjectures of Kononov--Pi--Shen.