20 problems

Completely reducible matrix-polynomial conjecture. This moduli space, consisting of completely reducible $n\times n$ degree $m$ matrix polynomials with spectral curve $C$, consider…

Maulik–Yun's conjecture. The perverse filtration and the Lefschetz filtration are opposite:

Galashin–Lam's conjecture. There exists a non-algebraic deformation retract

Multiplicative splitting conjecture. There exists such a splitting that is multiplicative under cup product:

Let $C_0$ be the central curve in the setting of Theorem 0.8, and let $\phi$ and $\phi'$ be nondegenerate stability conditions. Write $\overline{J}^{\phi}_{C_0}$ and…

Let $\bar J_{\mathcal C} \xrightarrow{\pi} B$ be a compactified Jacobian fibration associated to a flat family of integral projective locally planar curves of arithmetic genus $g$…

Let $m,n$ be coprime positive integers, let $C_{\frac{m}{n}}$ be the closure of $y^m=x^n$ in $\mathbb{P}^2$, and let $J_{\frac{m}{n}}$ be its compactified Jacobian. Let $P$ be the…

Let $\pi:\overline{J}_C\to B$ be a Lagrangian compactified Jacobian fibration as in the motivic Beauville decomposition conjecture, with motivic summands $h_{2i}(\overline{J}_C/B)$…

Let $\pi:\overline{J}_C\to B$ be as in the motivic Beauville decomposition conjecture, and let $\operatorname{Corr}^k_B(\overline{J}_C,\overline{J}_C)$ denote the degree-$k$ relati…

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…

Let $(r_1,s_1),\ldots,(r_{\ell},s_{\ell})$ be a sequence of characteristic pairs for an algebraic curve. Starting with $f_{\ell+1}:=p_1\in\Lambda$, recursively apply the operators…

Let $\Pit_{k,n}$ be the torus quotient and let $J_{k,n-k}$ be the compactified Jacobian of the plane curve singularity $x^k=y^{n-k}$. Their cohomology carries a weight filtration a…

Let $A$ be an Abelian surface of Picard rank one, let $\tau$ be the involution defining the quotient, and let $C\subset A$ be an integral $\tau$-invariant curve of geometric genus…

Let $\mathcal{C}/\mathcal{B}'$ be the family of curves and let $\mathcal{J}(\mathcal{C}/\mathcal{B}')$ be its relative compactified Jacobian. Let $h$ denote the projection to…

Let $C_{q/p}$ be the plane curve under consideration, let $J_{q/p}$ be its compactified Jacobian, and let $P$ be the perverse filtration on $\textup{H}^{*}(J_{q/p})$. Let…

Existence conjecture. There exists a sheaf $\mathcal I\in\overline J_{\mathrm{C}}$ such that

Let $C$ be a projective geometrically integral algebraic curve over $k$ whose singularities are all planar, and let $\overline{\operatorname{Jac}}_{C}$ be its compactified Jacobian…

Let $\mathcal A$ be the Hitchin base, let $H:M\to\mathcal A$ be the Hitchin fibration, and let $\overline P$ be the universal Poincaré sheaf from the global conjecture. For each…

For coprime positive integers $k,n$, let $L_{k/n}$ be the finite-dimensional simple representation of the rational Cherednik algebra with parameter $k/n$, let $eL_{k/n}$ be its sph…

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 geom…