I will introduce the P=C conjecture which is an analogue of the P=W conjecture in the setting of a del Pezzo surface S. The main character of the conjecture is the moduli space M of one-dimensional sheaves on S. The perverse filtration records the topology of the Hitchin-type fibration, while the Chern filtration records the shape of tautological relations. The perverse filtration is of particular interest because it conjecturally categorifies the Gopakumar-Vafa invariants of a local Calabi-Yau 3-fold. This is a joint work with Y. Kononov, M. Moreira, W. Pi.