40 problems

Let $V$ be a Fano manifold, and let stable mean stable in the sense of Geometric Invariant Theory. Yau's conjecture. The manifold $V$ admits a Kähler–Einstein metric if and only if…

Effective linearization conjecture.

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…

Let $\Sigma^\circ\subset |\mathcal O_{\mathbb P^9}(4)|$ be the locus of Coble type quartics singular exactly along a smooth hyperkähler fourfold in $\mathcal M_4^{(1)}$, and let…

Let $\mathcal{Y}$ be a smooth stack, let $\mathcal{S}\subset\mathcal{Y}$ be the relevant closed stratum, and let…

Let $Q$ be a quiver, let $d$ be an indivisible dimension vector such that … where $(d,e_i)$ is the relevant bilinear form and $Q_0$ is the vertex set. Let $\mathcal M_d^\chi(Q)$ be…

Let $n$ be a positive integer and let $\frac{1}{n}+\frac{}{}\epsilon$ be the common weight on $2n$ marked points, with the weighted moduli space…

Let $X$ be a monotone almost complex manifold with a Hamiltonian action of a group $G$, moment map $\mu$, and symplectic quotient $X/\!/G=\mu^{-1}(0)/G$. Let $\mathcal{F}(X)$ denot…

Let $Z$ be the affine bundle constructed from the extension data, let $G$ be the acting group, and let … be the resulting moment-map analogue. For $\zeta\in HH_0(\mathscr C)$, writ…

Let ${\cal K}$ and ${\cal H}_{\overline{y_e}}$ be the stabilizer-related groups associated with the orbit-closure data, and let $\overline{O(z)}\subseteq\overline{O(y)}$ be orbit c…

Gravitating coupled vortex conjecture. The pair $(\mathbb{P}^1,T)$ admits a solution to the gravitating coupled vortex equations if and only if $T$ is $\sigma$-polystable and the p…

Coherent completeness and formal functions conjecture. The stack $[X/G]$ is cohomologically proper over $A^G$, and the pair $([X/G],\mathcal{G}_x)$ is coherently complete and satis…

The algorithms use a Weyl group $W$ and a subset $W'\subset W$ in their final optimization step to remove nonmaximal states. Weyl group optimization conjecture. At the last step of…

Strong Tits Centre Conjecture for vector edifices. If $C$ is a closed convex non-completely-reducible cone in $V_G({\mathbb K})$, then $C$ has an unopposed $\Gamma$-centre.

Strong Tits Centre Conjecture. If $\Sigma$ is a closed convex non-completely-reducible subset of $\Delta_G({\mathbb K})$, then $\Sigma$ has an unopposed $\Gamma$-centre.

Tits Centre Conjecture. If $\Sigma$ is a convex non-completely-reducible subcomplex of $\Delta$, then $\Sigma$ has a simplicial $\Gamma$-centre.

Let $X$ be the smooth Fano threefold obtained by blowing up a smooth quadric threefold $Q\subset\mathbb{P}^4$ along the smooth twisted rational quartic curve $C_4$, so that…

Let $G$ be a reductive group acting on a smooth projective variety $A$, let $T$ be a maximal torus of $G$, and let $W$ be the Weyl group. Write $A /\!/ G$ and $A /\!/ T$ for the co…

Compatibility conjecture. The following diagram commutes:

Flag Hilbert scheme conjecture. The following hold:

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\operatorname{Hilb}(\mathfrak{g})$ denote its generalized Hilbert scheme. Generalized GIT quotient conjecture. There is a g…

Let $g$ and $p$ be integers, and let $\operatorname{Hilb}(\mathbb{P}^{g-1})_{p,2}^{\operatorname{ss}}$ denote the locus in the Hilbert scheme of genus $g$ canonically embedded curv…

Let $k$ be an integer and let $R_{2k+1}$ be the balanced canonical ribbon of genus $2k+1$. For integers $p$ satisfying $2\leq p\leq k-1$, consider its $(p,2)$-syzygy point…

Let $C\subset \mathbb{P}^{g-1}$ be a generic canonical curve. For integers $p$ with $p\geq \lfloor (g-3)/2\rfloor$, consider its $(p,2)$-syzygy point…

VGIT-ring conjecture. Under these assumptions, the image of the hyperkähler Kirwan map is equal to the VGIT ring: