57 problems

Simpson's standard conjecture. By spreading out, the $\overline{\mathbb{Q}}$-points in the intersection

Let $\mathcal{M}_B$ be a smooth Betti moduli space, or character variety, of complex dimension $d$ associated with a punctured Riemann surface, and let…

Compatibility conjecture. The isomorphisms between different moduli spaces given by the middle convolution map preserve the Hodge filtration, the ${\mathbb C}^{\ast}$ action, and t…

Nonabelian Hodge conjecture. The points of $NL(X_s,GL(n))$, that is, the $\mathbb Z$-variations of Hodge structure, are motivic representations on $X_s$.

Nonabelian Noether–Lefschetz algebraicity conjecture. If $S$ is a quasiprojective variety, then $NL(X/S,GL(n))$ is an algebraic variety and the morphisms

Kerz's commutativity conjecture. (i) For any $\lambda\in\mathbb R^*$, $\lambda\cdot\sigma_{\operatorname{Dol}}$ is also an isomonodromic deformation. (ii) For any…

Let $G$ be a semisimple group over $\a9\a9$ and let $G^{\mathrm{sc}}$ be its simply-connected cover. Let $\mathcal{M}^{\circ}_G$ denote the neutral component of the affine Toda mod…

Let $\mathfrak M_{(0,-2i\zeta_{\mathbb{R}})}$ be the indicated quiver-variety moduli space, let $p_0$ be a fixed point of the $\mathbb{C}^{\star}$-action, and let $W^1(p_0)$ be the…

Let $X$ be a smooth projective rigid analytic space over $\mathbb{C}_p$, and let $G$ be a reductive group defined over $\mathbb{C}_p$. A $G(\mathbb{C}_p)$-representation of…

Let $C$ be a smooth projective curve with underlying Riemann surface $\Sigma_g$. Let $\mathcal{H}^{\operatorname{Betti}}$ and $\mathcal{H}^{\operatorname{Dol}}$ be the graded BPS c…

Let $\alpha_S$ be the canonical homeomorphism between the relative de Rham and Higgs moduli spaces, and assume that it is an isomorphism of differentiable fiber bundles. Denote the…

Let $S$ be a smooth complex curve, let $X\to S$ be a polarized smooth family of smooth projective curves of genus at least $2$, and let $g:M^{s}_{Hig}(X/S,r)\to S$ be the relative…

Differentiable nonabelian Hodge conjecture. The map $\alpha_S$ is an isomorphism of differentiable fiber bundles over $S$. This would upgrade the known fiberwise topological identi…

Let $D$ be the boundary divisor in a compactification of the relevant character variety, and let the dual intersection complex of $D$ be the polyhedral complex encoding the interse…

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…

Let $G$ be a reductive Lie group, let $g$ be the genus of a closed Riemann surface, and let $\mathcal{X}_{g}(G)$ denote the corresponding character variety. A dlt log Calabi–Yau co…

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 $X$ be a smooth projective variety, let $r$ be a positive integer, and let $\mathsf{sd}_X:\mathcal{M}_{\mathrm{Higgs}}^{\mathrm{stack},r}\to \mathcal{S}_X^r$ be the spectral mo…

Let $X$ be the variety under consideration and let $\rho:\pi_1(X)\to \mathrm{GL}_r(\mathbb{C})$ be a rigid representation, meaning that its character-variety point is isolated. A r…

Let $\mathcal{M}_{B}(r,n)$ be the twisted character variety of a smooth projective curve and let $\mathcal{M}_{Dol}(r,n)$ be the moduli space of semistable Higgs bundles of rank…

Algebraicity conjecture. The space $\operatorname{Sh}_{\varrho}(X)$ is quasi-projective and $\operatorname{sh}_{\varrho}$ is an algebraic morphism.

A complex variation of Hodge structure is a geometric structure on a flat complex vector bundle whose fibers carry varying Hodge structures; a representation is rigid if it admits…

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 $(\overline{\partial}_E,\varphi)$ be a stable nilpotent Higgs bundle, and let $\Phi$ be the Higgs field associated with the leading term of the gauge-equivalent family of flat…