27 problems

Let $C$ be a smooth projective curve over $\mathbb{C}$, and let $D\subset C$ be a reduced effective divisor. A parabolic Higgs bundle over $(C,D)$ is called periodic when it is per…

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

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…

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

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

Let $\sf{H}_W$ be the Hecke category, let $\sf{AH}:\sf{H}_W^{\operatorname{op}}\to\mathsf{Mod}_2^b(\bb{A}_W)$ be the annular Hecke functor, and let $\bb{A}_{W,\epsilon}$ and…

Let $X$ be a compact Riemann surface, let $\Sigma$ be its underlying surface, and let $\mathrm{G}^\mathbb{R}$ be a simple Lie group. Write ${\mathcal M}(X,\mathrm{G}^\mathbb{R})$ for th…

Let $C$ be a smooth projective curve, and let $\mathrm{MHG}(C)$ and $\mathrm{PHG}(C)$ denote respectively the motivic and periodic parabolic Higgs categories from the source. Let ……

Let $C$ be a smooth projective complex curve, and let a periodic Higgs bundle on $C$ mean a Higgs bundle satisfying the source's periodicity condition. Polystability conjecture. An…

Let $C$ be a smooth projective algebraic curve over $\mathbb{C}$. A periodic parabolic Higgs/de Rham bundle means a periodic object of the corresponding parabolic Higgs or de Rham…

Let $C$ be a smooth projective curve. The grading functor $\operatorname{Gr}$ sends periodic parabolic de Rham bundles over $C$ to periodic parabolic Higgs bundles over $C$. Arithm…

Let $C$ and $D$ be as in the source's theorem on periodicity for motivic Higgs bundles, and let $C_{\mathrm{log}}$ denote the corresponding logarithmic curve. A periodic parabolic…

Let $g\geq 2$, let $\Sigma_g$ be a genus-$g$ surface, and let $p\colon \mathfrak{M}_{g,r}^{\operatorname{Betti}}\to \mathcal{M}_{g,r}^{\operatorname{Betti}}$ be the semisimplificat…

Let $G$) be a reductive group, and let $M_\mathrm{H}$ and $M_\mathrm{B}$ be the corresponding Hitchin and Betti moduli spaces for a curve $C$. Let $h:M_\mathrm{H}\to\mathbb{A}^d$ b…

Numerical $P=W$ conjecture. The perverse numbers of the Hitchin fibration and the Hodge numbers of the corresponding character variety should agree:

Let $X$ be a compact Riemann surface, let $D\subset X$ be a finite set of marked points, and fix corresponding parameters for filtered local systems and parabolic Higgs bundles of…

Let ${\rm M}_{Dol}(S,C_{\cdot})$ be the Dolbeault (Hitchin) moduli space and ${\rm M}_B(S,C_{\cdot})$ the Betti character variety. Let $N^{\ast}_{Dol}$ and $N^{\ast}_B$ be neighbor…

Twistor-family conjecture. The moduli space of stability conditions of a Fukaya–Seidel category can be included in a one-parameter twistor family.