146 problems

Let $x\in (X,\Delta,\mathbb{T})$ be a log Fano cone singularity over an uncountable algebraically closed field. For every $\mathbb{T}$-invariant valuation $\nu_0$ centered at $x$ w…

Let $X$ be a Fano variety with at most Kawamata log terminal singularities. An anti-canonical polar cylinder means a cylinder polarized by $-K_X$. The anti-canonical cylinder conje…

Let $(X,L)$ be a smooth polarized manifold, where $c_1(L)$ denotes the first Chern class of $L$, and let K-polystability mean that every test configuration has nonnegative Donaldso…

Let $n$ be a positive integer. A set of klt singularities is bounded if all its members occur as fibers of one $\mathbb{Q}$-Gorenstein family; for Fano cone singularities, the fami…

Let a polarized variety mean a projective variety equipped with a polarization, and consider the moduli space of K-polystable polarized varieties. K-moduli conjecture. A set of K-p…

Székelyhidi's conjecture. $X$ admits an extremal Kähler metric in the class of the polarization if and only if it is relatively K-polystable.

Delta-invariant conjecture. For every $n$, one has

Donaldson's conjecture. The following statements are equivalent: (1) there is no constant scalar curvature Kähler metric in $\mathcal H$; (2) there is a potential…

Donaldson's conjecture. There exists a solution of the modified Abreu equation in $\mathcal{S}(\Delta, {\bf L})$ if and only if $(\Delta, {\bf L}, f)$ is K-polystable.

Toric modified-Calabi-flow conjecture. The modified Calabi flow converges exponentially fast to an extremal metric in $c_1(L)$.

Let $F$ be a number field, $(X_{\eta},L_{\eta})$ a normal polarized projective variety over $F$, and let $(X,L,h_L)$ range over all metrized polarized normal models over…

Let $X$ be a Fano manifold. The K-stability–Gibbs stability conjecture. $X$ is (uniformly) K-stable if and only if $X$ is (uniformly) Gibbs stable. Here uniform Gibbs stability mea…

Birational superrigidity conjecture. A birationally superrigid Fano variety is K-stable.

Let a Fano fibration be a fibration whose fibres are Fano varieties, possibly with singular fibres in a degeneration. Fano-fibration degeneration conjecture. A Fano fibration is po…

Let $X$ be a normal projective variety, and let $D_j$ for $1\leq j\leq k$ be effective Weil divisors on $X$. The notation $\operatorname{Kss}(X,\sum_{j=1}^k D_j)$ denotes the K-sem…

The wall-crossing setup concerns a log-bounded family $\widetilde{\mathcal{G}}$ subject to a volume condition $(\clubsuit)$, under which Theorem … still holds without the volume co…

High-temperature uniqueness conjecture. For $\lambda\leq 0$, $\mu^\lambda$-cscK metrics are unique modulo the automorphism group.

Loewy filtration destabilisation conjecture. If $(X,L)$ has non-reductive automorphism group, then the Loewy filtration destabilises $(X,L)$.

Let $(X,L)$ be a polarised variety, let $T$ be a maximal torus of automorphisms of $X$, and let $F_\chi(\alpha)$ denote the modified Futaki invariant associated with the extremal v…

Let $(X,L)$ be a polarized manifold. Let $\mathcal{H}$ be the space of Kähler metrics in the class $c_1(L)$, let $\mathcal{H}^{\operatorname{NA}}$ be the space of smooth non-Archim…

Polarized semiclassical Yau–Tian–Donaldson conjecture. The following are equivalent:

Semiclassical Yau–Tian–Donaldson conjecture. The triple $(X,\sigma,\alpha)$ is Poisson K-polystable if and only if there exists $\epsilon>0$ such that, for every…

Jiang's conjecture. $\alpha(X)<\frac{1}{n}$ if and only if $X\cong \mathbb{P}^n$.

Let $(X,L)$ be a smooth polarized variety, and let $\mathbb{G}\subset \operatorname{Aut}^\circ(X,L)$ be a reductive complex Lie group. Let $(\mathcal{X},\mathcal{L})$ be a dominati…

Interpolation conjecture. For each $s\in[0,1]$, the ray $\{u_t^s\}$ is also approximable.