The cohomology of moduli spaces of 1-dimensional sheaves on del Pezzo surfaces carries a perverse filtration, from which curve counting invariants of CY 3-folds can be extracted. Conjecturally, the perverse filtration matches a second filtration, defined in terms of tautological classes, called the Chern filtration. I will explain this conjecture, what is currently known, and some consequences. In the second part of the talk I will explain some new ideas to prove properties of the Chern filtration by using parabolic sheaves. In particular, I will discuss the top Chern degree and chi-independence phenomena. The talk is based on joint work in progress with W. Lim and W. Pi.