22 problems

Let $(M,\omega,J)$ be a compact Kähler manifold, and let ${\mathcal K}$ denote the K-energy. Consider a $C^{1,1}$ minimizer of ${\mathcal K}$. Chen's regularity conjecture. Every s…

Let $(X,L)$ be a polarized smooth projective variety, where $L$ is an ample $\mathbb{Q}$-divisor. The pair $(X,L)$ is K-polystable when it satisfies the algebro-geometric K-polysta…

CSC–stability conjecture. The existence of a CSC Kähler metric in $\Omega$ should be equivalent to stability of the polarized projective variety $(M,L)$.

K-polystability conjecture. K-polystability for a polarized (Hodge) Kähler manifold $(M,J,\Omega)$ should be equivalent to the existence of a CSC Kähler metric in the Kähler class…

Let $(M,L)$ be a polarized Kähler manifold, meaning that $M$ is a compact complex manifold and $L$ is an ample line bundle, and let a cscK metric mean a Kähler metric of constant s…

JSS conjecture. If $X$ is a minimal model, then the cscK metric spaces near the canonical class $[K_X]$ converge geometrically to the twisted Kähler–Einstein space…

Let $(Y,H)$ be a polarised manifold, let $c_1(H)$ be its first Chern class, let $S(\omega)$ denote the scalar curvature of a Kähler metric $\omega\in c_1(H)$, and let $\hat S$ be t…

Let $(V,\omega,\mathbb{T})$ be a toric manifold and let $\mathrm{v}>0$ and $\mathrm{w}$ be weight functions. A $(\mathrm{v},\mathrm{w})$-csc almost Kähler metric is an involutive a…

Let $(V,\omega,\mathbb{T})$ be a toric manifold equipped with positive weight functions $\mathrm{v}$ and $\mathrm{w}$. A weighted involutive csc almost Kähler metric is an involuti…

Let $(X,L)$ be a polarized manifold equipped with the data defining a weighted scalar curvature and an associated notion of weighted K-stability. Weighted Yau–Tian–Donaldson conjec…

Let $Z$ be a compact Kähler manifold, let $L$ be a positive line bundle on $Z$, and let $\mathcal{M}$ denote the Mabuchi K-energy on the space $M_L$ of metrics on $L$. A generalize…

Let $(M,\omega,J)$ be a compact Kähler manifold, and let $K$ be a maximal compact subgroup of $\operatorname{Aut}_0(M)$. Let ${\mathcal H}$ be the space of Kähler potentials and le…

Let $X$ be a compact Kähler manifold, let $Z\subset X$ be a smooth divisor admitting a Kähler metric of non-positive constant scalar curvature, and let $L$ be a polarization. The t…

Uniform Yau--Tian--Donaldson conjecture. The class $c_1(L)$ contains a Kähler metric of constant scalar curvature if and only if $(X,L)$ is uniformly K-stable in the $L^1$-sense re…

Let $(M,[\omega],J)$ be a compact Kähler manifold and let $\mathcal E^1$ denote the energy class of $\omega$-plurisubharmonic potentials containing the metric completion of the smo…

Torus-relative stability conjecture. If

Let $M$ be a polarized manifold with polarization $L$, and consider Kähler metrics with Kähler class $2\pi c_1(L)$. Partial C^0-estimate conjecture. A version of the partial $C^0$-…

Let $(M,\xi)$ be a Sasakian manifold polarized by the Reeb vector field $\xi$, with polarized affine cone $(Y,\xi)$, where $Y=C(M)$ is the fixed complex cone. A constant scalar cur…

Let $X$ be a Kähler manifold or orbifold, and consider constant scalar curvature Kähler metrics on $X$. Such metrics need not exist. Stability conjecture. The existence of a consta…

Let $(X,J)$ be a polarized manifold, with $L$ an ample line bundle on $(X,J)$. A polarization $(X,J,L)$ is K-stable in the sense of algebraic K-stability, and $c_1(L)$ denotes the…

Let $(V,L)$ be a smooth polarised projective variety, where $c_1(L)$ is the first Chern class of the polarisation and a variety is $K$-stable in the sense of algebro-geometric stab…

Critical-point and convergence conjecture. If $\operatorname{Aut}_{0}(X)$ is trivial, then, for all sufficiently large $k$, $\mathcal{F}_{k}$ admits a critical point…