Perverse filtrations, Chern filtrations, and refined BPS invariants for local $\mathbb{P}^2$

TL;DR

This paper investigates the relationship between perverse and Chern filtrations in the cohomology of moduli spaces of sheaves on , proposing a conjecture linking them and providing partial proofs for low degrees.

Contribution

It formulates the P=C conjecture connecting perverse and Chern filtrations and proves it for degrees up to 4, advancing understanding of cohomological structures in enumerative geometry.

Findings

Formulated the P=C conjecture for local 

Proved the conjecture for degrees  

Connected filtrations with refined BPS invariants

Abstract

We explore connections between three structures associated with the cohomology of the moduli of 1-dimensional stable sheaves on $\mathbb{P}^2$: perverse filtrations, tautological classes, and refined BPS invariants for local $\mathbb{P}^2$. We formulate the $P=C$ conjecture identifying the perverse filtration with the Chern filtration for the free part of the cohomology. This can be viewed as an analog of de Cataldo--Hausel--Migliorini's $P=W$ conjecture for Hitchin systems. Our conjecture is compatible with the enumerative invariants of local $\mathbb{P}^2$ calculated by refined Pandharipande--Thomas theory or Nekrasov partition functions. It provides a cohomological lift of a conjectural product formula of the asymptotic refined BPS invariants. We prove the $P=C$ conjecture for degrees $\leq 4$.