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

We initiate a systematic study on the cohomology rings of the moduli stack $\mathfrak{M}_{d,\chi}$ of semistable one-dimensional sheaves on the projective plane. We introduce a set of tautological relations of geometric origin, including Mumford-type relations, and prove that their ideal is generated by certain primitive relations via the Virasoro operators. Using BPS integrality and the computational efficiency of Virasoro operators, we show that our geometric relations completely determine the cohomology rings of the moduli stacks up to degree 5. As an application, we verify the refined Gopakumar--Vafa/Pandharipande--Thomas correspondence for local $\mathbb{P}^2$ in degree 5. Furthermore, we propose a substantially strengthened version of the $P=C$ conjecture, originally introduced by Shen and two of the authors. This can be viewed as an analogue of the $P=W$ conjecture in a compact and Fano setting.