145 problems

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.

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 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…

Let $X$ be a Fano variety, and let $b3(X)$ denote the Gibbs stability invariant. Recall that K-stability and K-semistability are algebraic stability conditions for Fano varieties.…

Let $(\mathcal{X},\mathcal{D})$ be an arithmetic log pair over $\mathbb{Z}$ such that $-\mathcal{K}_{(\mathcal{X},\mathcal{D})}$ is a relatively ample $\mathbb{Q}$-line bundle, and…

Let $(X,\Delta)$ be a log Fano manifold with vanishing Futaki character. Define the divisorial polystability threshold $\delta(X,\Delta)_{\mathrm{ps}}$ as the infimum of the ratio…

Let $(X,\Delta)$ be a log Fano manifold with vanishing Futaki character. Let $\gamma(X,\Delta)^{\mathcal{G}}$ denote the reduced Gibbs stability threshold and let…

Let $(X,\Delta)$ be a log Fano manifold. Gibbs polystability conjecture. $(X,\Delta)$ is Gibbs polystable if and only if it is uniformly Gibbs polystable if and only if it is K-pol…

Let $M$ be a Fano manifold, meaning a compact complex manifold with positive first Chern class. A Fano manifold is K-stable when it satisfies the algebro-geometric K-stability cond…

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 $X$ be a smooth uniruled variety. K-stable-fibre conjecture. The variety $X$ is birational to a Mori fibre space whose general fibre is K-stable. This conjecture links biration…

Let $X$ be a K-semistable $\mathbb{Q}$-Gorenstein degeneration of a smooth Fano threefold of Family textnumero 2.16. Degeneration conjecture. $X$ is isomorphic to the blow-up of a…

Let $x\in X$ be an $n$-dimensional klt singularity, and let $\widehat{\operatorname{vol}}(x,X)$ denote its local volume, defined as the infimum of the normalized volumes of valuati…

Let $T$ be a torus and let $c>0$ be a real number. Consider isomorphism classes of $T$-equivariant $(\mathbb{Q})$-Fano fibrations $T\curvearrowright(X\xrightarrow{\pi}Y)$, with wei…

Quantized volume comparison conjecture. For every $m\geq 1$,