47 problems

de Cataldo–Hausel–Migliorini P=W conjecture. The diffeomorphism $\psi$ between the spaces respects these filtrations.

Let $X$ be a smooth Fano variety with a normal crossing anti-canonical complement $D=D_1+\ldots+D_r$, and let $U=X\setminus D$. Suppose that $U^\vee$ is a mirror dual open Calabi–Y…

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:

Kononov–Moreira–Lim–Pi conjecture. The associated graded group

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…

The P=W conjecture. Under this identification, the perverse filtration on $M_D$ associated with the Hitchin fibration matches the Hodge-theoretic weight filtration on $M_B$. This c…

Let $C$ be a nonsingular irreducible projective curve of genus $g\ge 2$. Let $M(n,d)$ be the coarse moduli space of rank $n$ degree $d$ semistable Higgs bundles on $C$, with Hitchi…

Let $M_{d,\chi}$ be the moduli space of one-dimensional sheaves on $\mathbb P^2$, with perverse filtration $P_\bullet H^*(M_{d,\chi})$ and tautological Chern filtration…

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 $f:M_{n,d}\to B$ be the Hitchin system, let $d_f=\dim M_{n,d}\times_B M_{n,d}-\dim M_{n,d}$, and let ${}^\mathfrak p\tau_{\leq k}$ denote perverse truncation. The perverse filt…

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…

Fix coprime integers $r,\chi$ with $r\geq 3$. Let $M_{r,\chi}$ be the moduli space of one-dimensional stable sheaves $F$ on $\mathbb P^2$ with $[\operatorname{supp}(F)]=rH$ and…

Let $M_{d,\chi}$ be the moduli space under consideration, and let $c_k(j)$ denote its tautological generators. The generators in the range $A^{*\leq d-2}(M_{d,\chi})$ are those lis…

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…

Let $r>0$ and $d$ be relatively prime integers, let $C$ be a smooth projective algebraic curve over $\mathbb C$, and let $M_{r,d}$ be the moduli space of stable Higgs bundles on…

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…

For elliptic $gamma$, let $cF_gamma$ denote the affine-Springer sheaf, and let $\operatorname{gr}^PcF_gamma$ denote the sheaf obtained from the perverse filtration. Perverse-filtra…