Cohomology rings of the moduli of one-dimensional sheaves on the projective plane

The moduli spaces of one-dimensional sheaves on the projective plane have been studied through their connections to enumerative geometry and representation theory. In this talk, I will explain a systematic approach to study their cohomology rings, using notably tautological relations of geometric origin. Our study leads to a conjecture that describes a highly nontrivial perverse filtration (which carries important enumerative data) on the cohomology in terms of explicit ring generators. This can be viewed as an analogue of the P=W conjecture in a compact and Fano setting. Based on joint work with Y. Kononov, W. Lim, M. Moreira, and J. Shen.