30 problems

Consider the generating series … Set … where $\mathsf{S}^\bullet$ denotes the plethystic exponential, and let $[H(q,t)]^{i,j}$ denote the coefficient of $q^it^j$ in the expansion a…

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

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

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 $g$…

Local-to-global multiplicativity conjecture. The perverse filtration associated with $f$ is multiplicative if and only if the local perverse filtrations on all fibers are multiplic…

Let $X$ be a compact hyperkähler manifold, and define … The perverse-Hodge octahedron conjecture. The convex hull of the non-zero $h^{i,k,d}$ is an octahedron. This is stronger tha…

Multiplicative motivic perverse filtration conjecture. The motivic perverse filtration is multiplicative with respect to cup-product:

Let $S$ be a del Pezzo surface, let $H$ be the polarization, let $\beta$ be an effective divisor, let $\chi\in\mathbb{Z}$, and let $M_{\beta,\chi}$ be the moduli space of one-dimen…

Let $M_{d,\chi}$ be the moduli space associated with the curve class $\beta=dH$ and Euler characteristic $\chi\in\mathbb{Z}$ on $\mathbb{P}^2$, where $H$ is the class of a line and…

Let $M_{d,c7}$ be the moduli space of one-dimensional sheaves on $\mathbb{P}^2$ with degree $d$ and Euler characteristic $\chi$, and let $P_\bullet H^*(M_{d,\chi},\mathbb{Q})$ and…

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…

Kononov–Pi–Shen's P=C conjecture. For $d\geq 3$, we have

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 $M_{d,\chi}$ be the moduli space considered in the paper, with perverse filtration $P_k$ on its cohomology. For $d\geq 3$, let $H^{*\leq 2d-4}(M_{d,\chi},\mathbb{Q})$ be the fr…

de Cataldo–Hausel–Migliorini P=W conjecture. Under this identification, the perverse filtration associated with $\pi$ satisfies

Let $U$ and $U^\vee$ be mirror $n$-dimensional log Calabi–Yau varieties. For a nonsingular quasi-projective variety $V$, let … be its perverse-mixed Hodge polynomial. Mirror P=W co…

Let $G=GL_n$, let $\gamma$ be the regular element used to define the affine Springer fiber $\Fl^{\gamma}_{\bI}$, let $\Lambda$ be the lattice acting on it, and let $P$ be the perve…

Let $C$ be a punctured Riemann surface, let $G$ be a linear reductive algebraic group over $\mathbb{C}$, and let $\mathcal{M}_B$ and $\mathcal{M}_{Dol}$ be corresponding Betti and…

Integer-finite local-system conjecture. Then $L$ is an integer finite local system, and there is an isomorphism

Perverse topological mirror symmetry.

P=W for resolution. The lift satisfies, for every $k$,

P=W conjecture for untwisted character varieties. For every $k$,

P=W conjecture for twisted character varieties. For every $k$,

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 $X$ be a compact Riemann surface, let $G$ be a complex reductive algebraic group, and choose resolutions of singularities … Let…