100 problems

Let $X$ be a compact Riemann surface and let $(E,\Phi)$ be a stable $\mathrm{SL}(n,\mathbb{C})$ Higgs bundle on $X$. For each $r>0$, let $e_r(x)$ denote the energy density at…

For every smooth projective complex variety $X$ and reductive algebraic group $G$, let $h_{\mathrm{sp}}:\mathcal{M}_{G\text{-Higgs}}(X)\to\mathcal{A}_{\mathrm{sp}}(X,G)$ be the spe…

Chern-class vanishing conjecture. All Chern classes of $\mathfrak{E}$ vanish.

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…

Let $G=\operatorname{GL}_r$. Let $\mathcal{B}\subset(\mathrm{B}_{\operatorname{GL}_r}^{\mathrm{red}})^{\mathrm{cl}}$ be an open subset, and set … Let…

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, let $X$ be an irreducible smooth projective variety over $\mathbb{K}$, and let $G$ be a reductive group ove…

Gradedness conjecture. Then $(E,\theta)$ is graded.

Let $\mathcal{X}$ be a stacky curve, let $\alpha$ be a fixed parabolic structure, and let $\widetilde{\mathcal{M}}^{ss}_H(\mathcal{X},\alpha)$ denote the moduli stack of semistable…

Let ${\cal M}$ be Hitchin's moduli space of rank-$2$ Higgs bundles with fixed determinant of degree $1$ over a Riemann surface $\Sigma$ of genus $g>1$. It is a simply connected, no…

The geometric Langlands correspondence. There exists a canonical equivalence of categories

Let $\mathcal{M}$ be the moduli space above, let $H_I^*(\mathcal{M})\subset H^*(\mathcal{M})^\Gamma$ be the subring generated by the universal classes $\alpha_{\mathcal{M}}$,…

Let $G$ be a complex connected reductive group, and let $M_{Dol}^{vc}(G)$ be the moduli space of $G$-Higgs bundles with vanishing Chern classes over a compact Riemann surface of ge…

Let $G$ be a reductive group, let $\chi\in\pi_1(G)$ and let $w\in Z_G^{\vee}$, where $Z_G$ is the center of $G$. Let $\mathrm{Higgs}_G(\chi)$ denote the connected component of the…

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 $C$ be a smooth projective curve, let $G$ be a reductive group with Langlands dual group $G^{\vee}$, and let $w$ denote the relevant stratum or weight. Write…

Bruzzo's conjecture. The following conditions are equivalent:

Let $W^1(\bar\partial_0,\Phi_0)$ be one of the Lagrangians arising from the conformal-limit construction, and let $M_{dR}$ be the de Rham moduli space. Simpson's conjecture. The La…

Let $\mathbb L_H(d)$ be the Lagrangian correspondence constructed between the relevant Hitchin moduli space and the Hilbert scheme of points on $T^*C$, and let the Dolbeault geomet…

Chen–Ngô's conjecture. For every point $s\in\mathscr{B}^r_X$, the fiber

Let $p_i$ be punctures with parabolic weights $\boldsymbol{\alpha}=(\alpha_i)$, and call the flags not all full when, for some $p_i$, the corresponding weight satisfies…

Bruzzo and Graña Otero conjecture. The Higgs bundle $\mathfrak{E}$ is semistable with respect to some polarization $H$ and $\Delta(E)=0$.

Let $\mathcal T_g$ denote the Teichmüller space of compact Riemann surfaces of genus $g\geq2$, let $\mathcal X/\mathcal T_g$ be its universal family, and let…

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…

Let $C$ be a smooth projective curve and let $G$ be a semisimple group, with Langlands dual group $G^{\vee}$. Let…

Let $C$ be a smooth projective curve with $g(C)\geq 2$, let $G$ be a connected semisimple group, and let $G^{\vee}$ be its Langlands dual. Let ${\mathcal L}$ be a line bundle on…