185 problems

Let $X$ be a smooth algebraic curve of genus $g\ge 2$, let $L\in\operatorname{Pic}(X)$ have degree $1$, and let $\Gamma=\operatorname{Pic}^0(X)[n]$. Denote by … the moduli space of…

Gaiotto–Moore–Neitzke's conjecture. As the radial variable $t\to\infty$ on the Hitchin base, the Hitchin metric on $\mathcal{M}$ is asymptotic to the semi-flat metric constructed f…

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

Uniruledness conjecture. The Kodaira dimension of $S$ is $-\infty$. In particular, if $S$ is a projective manifold, then $S$ is uniruled.

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…

Let $X$ be a projective smooth curve over $\mathbf C=\mathbf C_p$, the complete algebraic closure of $\mathbf Q_p$. A semistable Higgs bundle on $X$ is a Higgs bundle of degree $0$…

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

Mozgovoy's motivic ADHM conjecture. For every $r\geq 1$, $H_r(t)$ is a polynomial in $t$ and

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…

Let $X$ be a smooth projective curve over $\mathbb C$, let $G$ be a reductive group, and let $\mathrm{Higgs}_G(X)$ denote the moduli stack of $G$-Higgs bundles on $X$. Let…

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 $G^\textbf{R}$ be a real form of a complex simple Lie group, and let $\hat{\rho}$ be a magical $\mathfrak{sl}_2$-triple of $G^\textbf{R}$. The generalized Cayley correspondence…

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

Let $M$ be a two-dimensional moduli space of meromorphic Higgs bundles, and let $n$ be a positive integer. Boalch's conjecture. The Hilbert scheme of $n$ points on $M$ is itself a…

Let $\mathcal{N}$ be the moduli space of stable vector bundles, and let $\mathcal{W}$ denote the locus of wobbly bundles, namely stable bundles that are not very stable. A bundle i…

Viehweg–Zuo's conjecture. Any smooth fiber of a non-rigid family of hypersurfaces in $\mathbb P^3$ of degree $d$ is defined, up to a projective transformation, by an equation

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…

Surjectivity conjecture. The spectral data morphism is surjective. This predicts that the postulated image is exactly attained by Higgs bundles, although the source excerpt does no…

Hitchin WKB conjecture. As $t\to\infty$, for every non-critical path $\gamma$,

Let $G$ be a complex reductive group with Langlands dual group $G^L$, and let $\mathcal H_G$ and $\mathcal H_{G^L}$ be their Higgs moduli spaces. Let $\sigma_G$ be an anti-holomorp…

Gaiotto–Witten bijectivity conjecture. The maps $I_{\mathrm{NP}}$ and $I_{\mathrm{GNP}}$ are both bijective. This is the proposed Kobayashi–Hitchin-type correspondence for the exte…

Let $(E,\theta)$ be a nilpotent semistable Higgs bundle of exponent at most $p$. Strong Higgs semistability conjecture. The Higgs bundle $(E,\theta)$ is strongly Higgs semistable.…

Solubility conjecture. Every small ${\mathscr O}_{\mathfrak X}[\frac 1 p]$-Higgs bundle with coefficients in $\xi^{-1}\Omega^1_{{\mathfrak X}/{\mathscr S}}$ is soluble. This is par…