21 problems

Let $G$ be a reductive group, let $C$ be a curve, and let $E$ be a principal $G$-bundle over $C$ with canonical parabolic reduction. Behrend's conjecture. The canonical parabolic r…

Let $C$ be a smooth projective curve over an algebraically closed field of characteristic $p$, and consider the moduli space of semistable parabolic bundles of rank $2$ on $C$ with…

Trace formula conjecture. One has

Degree-independence conjecture. For every integer $d$, the sum

Let $X$ be a compact Riemann surface, let $P\subset X$ be a finite set, and let $(V,\nabla_V)$ be a semisimple holomorphic flat bundle on $X\setminus P$. Let $(H,D_V,h)$ be the tam…

Parabolic stability necessity conjecture. If $\widehat M$ admits a scalar-flat Kähler metric, then $E$ must be parabolically stable. This is a necessary-condition conjecture for th…

LeBrun–Simanca conjecture. The Kähler class $[\omega]$ contains a scalar-flat Kähler representative if and only if the parabolic bundle $E$ is stable. This conjectures that the qua…

Let $N$ and $s$ be integers with $0<s<N$, and let … For $\alpha\in W(N,s)^{\circ}$, let $M(\underline{\mathbf{1}})^\alpha$ be the moduli scheme of quasiparabolic bundles with multi…

Kostov's conjecture. There is an irreducible solution to

Let $M$ be a blown-up ruled surface of genus at least $2$, and let $[\omega]$ be an admissible Kähler class, meaning that $c_1\cup[\omega]=0$. Associate to $[\omega]$ its correspon…

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ satisfy the stated hypotheses, fix a parabolic subgroup $P$, and let $\mathcal{E}$ be an augmented commutative…

Let $d53d_\alpha(d,\{\overrightarrow{d^{(i)}}\}_{i=1,\cdots,n},\{\overrightarrow{w^{(i)}}\}_{i=1,\cdots,n})$ be a connected component of the locus of $\mathbb{C}^*$-fixed points in…

Periodic de Rham conjecture.

Weak periodic de Rham conjecture. The natural inclusion functor

Extension conjecture. The canonical parabolic extension of an irreducible periodic de Rham bundle over $U$ is again periodic.

Let ${\cal H}_p(S^2,n)$ be the space of Hecke modifications of stable rank $2$ parabolic bundles over $\mathbbm{CP}^1$ with three marked points, and let $M^{ss}(S^2,n+3)$ be the mo…

The cscK cone metric and log-K-polystability conjecture. The following three conditions are equivalent: (i) $X$ admits a cscK cone metric along $\mathcal{D}$ in any ample class

General geometric conjecture. There exists an action of $\mathfrak{H}$ on $H_c^i(\mathcal{X}_{\bf S}^{\mathcal{E}},\mathbb{C})$ such that

Geometric conjecture. (i) The quotient exists as an algebraic variety and $X_{\bf A}^{\mathcal{E}}\to\mathcal{X}_{\bf A}^{\mathcal{E}}$ is a principal…

Deligne's conjecture. There exists a polynomial $A^{{\bf b};\mathbf{m}}_{\boldsymbol{\mu}}(t)\in\mathcal{C}h(S_{\boldsymbol{\mu}})[t]$, with coefficients in the positive cone…

Let $E^{min}$ be a parabolic extension attaining the minimum of the local parabolic invariant. There might be several such minimal extensions, although the uniqueness conjecture. u…