19 problems

Birational invariance conjecture. The set-theoretic image $\operatorname{im}(h^r_X)$ of the Hitchin morphism is invariant under smooth birational modifications.

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

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…

The open-embedding conjecture. The morphism $\iota_\mathfrak{c}$ is an open embedding. This conjecture concerns the relation between the regular centralizer group scheme $J$ and th…

Let $\mathcal{M}_{B}(r,n)$ and $\mathcal{M}_{Dol}(r,n)$ be the Betti and Dolbeault moduli spaces, let $h$ be the Hitchin map, and let $W_\bullet$ and $P_\bullet$ denote the weight…

Let $\mathcal{M}_{B}^d$ be the Betti moduli space of rank-$n$ representations with one puncture and central monodromy, and let $\mathcal{M}_{Dol}^d$ be the corresponding Dolbeault…

Let $S$ be a closed surface, let $\hat{\rho}$ be a Hitchin representation, and let $\rho\in \mathcal M_{\text{Betti}}(S)$ satisfy … For a Riemann surface structure…

Let $\Sigma$ be a Riemann surface and let $\mathcal M_{\text{Higgs}}(\Sigma)$ be the moduli space of Higgs bundles, equipped with its Hitchin fibration. A Hitchin section is the se…

Conjecture. The union

Let $X$ be an algebraic surface, and let $\mathscr B_X^{\Diamond}(k)$ be the distinguished subset of the postulated Hitchin base $\mathscr B_X(k)$ defined in the surrounding discus…

Let $X$ be a proper smooth algebraic variety over an algebraically closed field $k$ of characteristic zero, let $\mathcal{M}_X$ be the moduli stack of Higgs bundles, and let…

Let $G$ be a reductive group with Lie algebra ${\mathfrak g}$, let ${\mathfrak t}$ be a Cartan subalgebra, let $W$ be the Weyl group, and let ${\mathfrak C}^d_G$ be the scheme of c…

Dai–Li's conjecture. The energy densities satisfy

Deroin–Tholozan conjecture. In each Hitchin fiber, the Hitchin section maximizes the translation length.

Explicit Langlands dual brane conjecture. The $(B,A,A)$-brane has a dual $(B,B,B)$-brane whose support is as follows. For odd $q$, it is

Let $G$ be a quasi-split group in the generality considered in the paper, let $a\in\mathfrak{c}(F)^{\textup{rs}}_{\nu}$, and use the perverse and Chern filtrations, the affine Spri…

Connectivity conjecture. Every connected component of the fixed-point set of $\Theta_{G}$ intersects the generic fibre of this Hitchin fibration.

Let $\mathcal A$ be the Hitchin base, let $H:M\to\mathcal A$ be the Hitchin fibration, and let $\overline P$ be the universal Poincaré sheaf from the global conjecture. For each…

Let $C$ be the underlying smooth projective curve, let $r$ and $d$ be the rank and degree, let $M=\mathcal M(d,r,\omega_C)$ be the moduli space of $S$-equivalence classes of Higgs…