The perverse–Lefschetz conjecture for compactified Jacobians

Let $\mathcal{C}\to\mathcal{B}$ be a family satisfying assumptions (A1)–(A4), and let $\overline{\mathcal{J}}\to\mathcal{B}$ be its family of compactified Jacobians. For every geometric point $b\in\mathcal{B}$, write $P_{\leq i}$ for the perverse filtration and $F^{\geq i}$ for the Lefschetz filtration on $\operatorname{H}^{*}(\overline{\mathcal{J}}_{b})$. Two filtrations are opposite when their associated graded pieces give a direct-sum decomposition of the vector space, equivalently when $\operatorname{H}^{*}(\overline{\mathcal{J}}_{b})=P_{\leq i}\oplus F^{\geq i+1}$ for every $i$. The perverse–Lefschetz conjecture. Assume (A1)–(A4) hold for $\mathcal{C}\to\mathcal{B}$. Then for every geometric point $b\in\mathcal{B}$, the perverse filtrations $P_{\leq i}$ and the Lefschetz filtration $F^{\geq i}$ on $\operatorname{H}^{*}(\overline{\mathcal{J}}_{b})$ are opposite to each other. The conjecture proposes that the perverse filtration arising from the family and the Lefschetz filtration induced by the determinant line bundle are complementary on every fiber, extending the agreement suggested by relative hard Lefschetz beyond the smooth locus. Its resolution is not supplied in the given text.