89 problems

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold, and let $[\omega]$ be a Kähler class. A Kähler metric in $[\omega]$ is constant scalar curvature (cscK) when its scalar curv…

Yau's conjecture. The geometric-invariant-theoretic condition should be equivalent to the existence of a Kähler–Einstein metric on $M$.

Let $M$ be a compact Kähler manifold with positive first Chern class, so that $M$ is Fano. A Kähler–Einstein metric on $M$ is a Kähler metric whose Ricci curvature is proportional…

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

Mumford's conjecture. Every smooth affine geometrically reductive group scheme over $k$ is reductive, and every smooth affine reductive group scheme over $k$ is geometrically reduc…

Kaledin–Lehn–Sorger conjecture. If there exists a symplectic resolution of

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 $M$ be a $\mathbb Q$-Fano group compactification, and consider the Kähler-Ricci flow on $M$ with initial metric $\omega_0$. Suppose its limit $M_\infty$ is obtained through two…

Flag Hilbert scheme conjecture. The following hold:

A canonical metric is a metric associated naturally with a geometric structure, while stability refers to a suitable notion of stability in geometric invariant theory. Yau's conjec…

Guillemin–Sternberg's conjecture. The equivariant holomorphic Euler characteristics should agree:

Let $G=K\exp\mathfrak p$ be a compatible subgroup of a complex Lie group acting holomorphically on a space $Z$, with moment map and associated semistable set…

Białynicki-Birula–Sommese conjecture. Part of the algebraic theory of GIT quotients should extend to the more general case of meromorphic actions of $({\mathbb C}^*)^k$ on a reduce…

Nonabelian/abelian correspondence. The genus zero Gromov–Witten invariants of $X//G$ are expressible in terms of genus zero Gromov–Witten invariants of $X//T$ twisted by the Euler…

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 $G$ be a connected reductive algebraic group over $\mathbb{Q}_p$, and let ${\mathcal F}$ be the homogeneous space of filtrations on an $F$-isocrystal with $G$-structure. Write…

Tian's conjecture. Hilbert–Mumford stability of $X_f$ is sufficient for the existence of a Kähler–Einstein metric on $X_f$.

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…

Main conjecture. (i) The map

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

Let $\mathscr{X}\rightarrow U$ be the family of Fano threefolds of Family textnumero 2.16, let $\lambda_{\operatorname{CM}}$ be its CM line bundle, and let $U^{\operatorname{ss}}$…

Geometric-window inequalities conjecture. The Chern–Simons levels should satisfy, in general,

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…

Mulmuley's polynomial-time conjecture. Orbit closure intersection problems for arbitrary connected reductive groups are in $\textnormal{\textbf{P}}$.