40 problems

Let $\mathcal{C}$ be a generic tuple of conjugacy classes, let $\bs\mu$ encode the eigenvalue multiplicities, and let $\mathcal{M}_{\mathcal{C}}$ be the associated character variet…

Let ${\mathcal{M}}_\B^{{\bm \mu},{\bf r}}$ be the generic wild character variety, let $d_{{\bm \mu},{\bf r}}$ be its dimension, and let…

Let $\mathcal{M}_{\mathcal{C}}$ be a character variety with its mixed Hodge structure, and suppose its mixed Hodge polynomial has the curious Poincare-duality symmetry described in…

Let $X$ be a character variety, and let $h^{j,p,q}(X)$ denote its Hodge numbers. Hausel's Hodge–Tate conjecture. The cohomologies of character varieties are Hodge–Tate: all Hodge n…

Polynomiality and positivity conjectures. The following two assertions are conjectured:

Hausel–Letellier–Villegas' conjecture. The mixed Hodge polynomials of character varieties of Riemann surfaces with semisimple conjugacy classes at the punctures are governed by Mac…

Genus-one partition identity. The following combinatorial identity holds:

Let $\mathbb{O}$ be a nilpotent orbit of a complex group, and consider the filtration of irreducible Harish-Chandra modules arising from quantization, together with the Hodge filtr…

For integers $g\geq 1$, $n\geq 0$, and $d\geq 0$, let $\mathcal{M}_{g,n}(\mathbb{P}^r,d)$ denote the moduli stack of stable maps to projective $r$-space, and let its compactly supp…

Kostka-weighted formula. The polynomial $C^f_{m,n}(q,t)$ coincides with the bigraded polynomial arising from either the mixed Hodge filtration or the perverse filtration of the Hit…

Fix the number $k$ of marked points and the genus $g$. For partitions $\mu_1,\ldots,\mu_k$ and the corresponding generic character variety $X_{g;J_{\mu_1},\ldots,J_{\mu_k}}$, let…

Let $X_B$ be the Betti character variety and let $H^*_{\mathrm{pure}}(X_B)=\bigoplus_i W_iH^i(X_B)$ denote its pure cohomology, where $W_\bullet$ is the weight filtration. Let…

Let $\Psi$ be the non-abelian Hodge correspondence between the Dolbeault moduli space and the character variety. The perverse filtration is the filtration arising from the relevant…

Mixed-Hodge-polynomial conjecture. The total number of fixed manifolds of $\mathcal{M}_{g,n}$, equivalently the number of simple modules of $V_{g,n}$, is

Let $C_{g,n}$ be the character variety of the $SU(2)/\mathbb{Z}_2$ Hitchin system on the punctured surface, and let $PH_{g,n}(q)$ be its pure mixed Hodge polynomial. Define the ren…

Libgober's conjecture. If $X$ is rationally elliptic, then its mixed Hodge structure is of Hodge–Tate type; equivalently, every non-zero mixed Hodge number $h^{k,p,q}$ satisfies

Hausel–Lettelier–Rodriguez-Villegas conjecture. The variety $\mathcal{M}_n$ has the same mixed Hodge polynomial as $\mathcal{M}_1^{[n]}$, and their $E$-polynomials agree.

Let $\mathcal{C}$ be a $k$-tuple of semisimple orbits, let $\alpha_{\mathcal{C}}$ be its associated dimension vector, and let…

Let $\mathcal{C}$ be a generic $k$-tuple of semisimple conjugacy classes, let $\alpha_{\mathcal{C}}$ be its associated dimension vector, and let…

Let $\mathcal{M}_{\mu,\nu}$ be the partial resolution of the character variety defined in the source, and let $w$ be an $n$-cycle in the Weyl group relative to the third puncture.…

Let $\mathcal{C}=(C_j)_{1\le j\le 2k}$ be a strongly generic tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$ satisfying the condition referred to…

Let $C$ be a nonsingular irreducible projective curve of genus $g\geq 2$, and let $n$ and $d$ be coprime integers. Let $M_{\mathrm{Dol}}(n,d)$ be the moduli space of stable Higgs b…

Let $U$ be a log Calabi–Yau variety and assume that its homological mirror $U^\vee$ is also a log Calabi–Yau variety with the same dimension. Set $d=\dim U=\dim U^\vee$. The perver…

For a partition $lambda$, let $\mathcal{H}_{lambda}(z,w)$ be the genus-one hook function, and let $lvertlambdarvert$ denote the size of $lambda$. Hausel–Rodriguez-Villegas genus-on…

Let $\mathcal{M}_n$ be the moduli space under consideration, let $d_n$ denote its dimension, and let $\mathcal{H}_{lambda}(z,w)$, $\overline{H}_n(z,w)$, and $H(\mathcal{M}_n;q,t)$…