14 problems

Let $G$ be a reductive group with Langlands dual $G^\vee$, let $C$ be the underlying curve, and let $D=\{c_i\}$ be the parabolic divisor. Let $(E,\varphi,Q)$ be a smooth fixed poin…

General double star-shaped extension conjecture. The same result holds for hyperpolygons with more general double star-shaped quivers and strongly parabolic Higgs bundles on…

Let $G=\operatorname{SO}_{2n}$, let $\mathbf{H}_{\overline{\mathbf{O}}}$ be the Hitchin base, and let $\mathbf{A}_P$ denote the Coulomb Hitchin base associated with a parabolic sub…

Let $Y$ be the spectral cover in the logarithmic cotangent bundle $T^{\vee}X(\log {\rm Wob})$, let ${\rm Wob}$ be the wobbly locus, and suppose a logarithmic Higgs sheaf is equippe…

Let $G$ be a group, let $C$ be a curve, and let $\Lambda_{dR}$ be an ${}^LG$ local system on $C$ corresponding to a point $\Lambda_{Dol}$ in the Hitchin fiber of…

Purity conjecture. When $n>1$, the $\mathbb{Q}$-mixed Hodge structure on $H^*(\mathcal{H}_\xi)$ is of pure type for every $\xi \in \mathcal{N}^\circ$.

Let $C$ be a smooth projective curve of genus $g$, let $p_1,\dots,p_N$ be distinct $k$-rational points, and write $D=p_1+\cdots+p_N$. For a weight $\alpha$, let…

Let $X$ be a compact Riemann surface with marked points and fixed parabolic parameters, and let the corresponding character variety parametrize representations of the fundamental g…

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 ${\mathcal{M}}_\Dol^{\bm \mu}$ be the parabolic Higgs moduli space and ${\mathcal{M}}_\B^{\bm \mu}$ the corresponding character variety, related by non-abelian Hodge theory. Th…

Let $C$ be a complex smooth projective curve of genus $g$, with divisor $D=p_1+\dots+p_k+rp$, where the points are distinct and $p$ has multiplicity $r$. Let ${\bm \mu}$ be a $k$-t…

Let $C$ be a complex smooth projective curve, let $D=p_1+cdocdots+p_k+rp$, and let ${\mathcal{M}}_\Dol^{{\bm \mu},r}$ be the corresponding moduli space of stable twisted parabolic…

Consider the character varieties associated with the five families of parabolic Higgs-bundle moduli spaces in the paper, with monodromy around the punctures. The Hodge-Tate conject…

Let ${\mathcal C}^e_{\underline\lambda}$ be the relevant character variety and let ${\mathcal H}^e_{\underline m}$ be the moduli space of stable parabolic Higgs bundles with primit…