On the perverse filtration of the moduli spaces of 1-dimensional sheaves on ${\mathbb{P}}^2$ and P=C conjecture

Let $M(d,\chi )$ be the moduli space of semistable 1-dimensional sheaves supported at curves of degree $d$ on ${\mathbb{P}}^2$, with Euler characteristic $\chi$. We have the Hilbert-Chow morphism $\pi :M(d,\chi )\to \mathrm{\mid }dH\mathrm{\mid }$ sending each sheaf to its support. We study the perverse filtration on $H^{\ast }(M(d,\chi ),\mathbb{Q})$ via map $\pi$, especially the P=C conjecture posed by Kononov-Pi-Shen. We show that P=C conjecture holds for $H^{\ast \le 4}(M(d,\chi ),\mathbb{Q})$ for any $d\ge 4$, $(d,\chi )=1$. The main strategy is to relate $M(d,\chi )$ to the Hilbert scheme $S^{[n]}$ of $n$-points and transfer the problem to some properties on $H^{\ast }(S^{[n]},\mathbb{Q})$. We use induction on $n$ to achieve the desired properties. Our proof involves some complicated calculations.