The perverse–weight correspondence conjecture for compactified Picards

Let $C$ be a complex integral projective curve whose normalization has genus zero and whose unique planar singularity is locally given by $f(x,y)=0$. Let $\mathcal{C}\operatorname{ptPic}/\Lambda$ be the compactified Picard quotient, equipped with the perverse filtration $P$ and weight filtration $W$ on cohomology, and let $Q$ denote the filtration induced by the Quot-side description. Perverse–weight correspondence conjecture. The weight grading on $\operatorname{gr}^{W}_*H^*(\mathcal{C}\operatorname{ptPic}/\Lambda)$ is supported in even degrees. Moreover, for all $j,k$,

$$\operatorname{gr}^{P}_{j+k}\operatorname{gr}^{W}_{2k}H^*(\mathcal{C}\operatorname{ptPic}/\Lambda)\simeq\operatorname{gr}^{Q}_{j}\operatorname{gr}^{W}_{2k}H^*(\mathcal{C}\operatorname{ptPic}/\Lambda).$$

This strengthens Hilb-vs-Quot by identifying the perverse and Quot filtrations after a weight-dependent shift, and extends an unpublished conjecture of Yun beyond the unibranch case. Its general status is open.