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

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 $\le 4$.