57 problems

ODP conjecture.

Let $X$ be a Fano manifold. A Kähler–Einstein metric on $X$ is denoted by $\omega_{KE}$, and let $\delta_N$ be the empirical measure of the corresponding canonical point process. T…

Picard number conjecture. Then $S$ has Picard number one.

Let $\pi:\mathcal{X}\rightarrow\Delta$ be a smoothing of a Kähler–Einstein orbifold $(X_0,\omega_0)$ with $A_k$-singularities, and suppose that $\operatorname{Aut}(X_0)$ is discret…

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…

Let $\Sigma$ be the closed surface under consideration, let $\mathcal L^*(g)$ denote the relevant set of surface subgroups of genus $g$, and let $A(h)(\Pi)$ be the associated area…

Higher-dimensional coupled Kähler–Einstein conjecture. For every integer $n\geq 4$, there exists a toric Fano manifold of dimension $n$ that admits a two-coupled Kähler–Einstein me…

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…

Folklore monotonicity conjecture. There exists $k_0\in\mathbb{N}_+$ such that, whenever $k_0\leq k_1\leq k_2$,

Tian's stabilization conjecture. There exists $k_0\in\mathbb{N}_+$ such that, for every $k\geq k_0$,

Let $X$ be an $n$-dimensional normal compact Kähler space, and let $\omega_{KE}$ be a singular Kähler–Einstein metric on $X$ satisfying … on the regular set $X^{reg}$, with induced…

Let $(S,C)$ be a strongly asymptotically log del Pezzo surface with $r=1$, where $S$ is a smooth surface and $C$ is a smooth irreducible curve. The notation $(K_S+C)^2$ denotes the…

Let $(X_j,\omega_j,p_j)$ be a sequence of $n$-dimensional Kähler–Einstein manifolds with $\operatorname{Ric}(\omega_j)=c_j\omega_j$, $|c_j|\leq 1$, and…

An $S^1$-invariant Kähler manifold is a Kähler manifold admitting, in suitable local holomorphic coordinates, a Kähler potential depending only on $(|z_1|^2,\dots,|z_n|^2)$. A Kähl…

A Kähler–Einstein manifold is a Kähler manifold whose Ricci tensor is proportional to its metric, and a Kähler immersion is a holomorphic isometric immersion into a complex space f…

Let $(\mathcal{X},\mathcal{L})$ be an arithmetic variety as above, and let $X$ be its corresponding Fano manifold. Assume that $X$ admits a unique Kähler–Einstein metric, with volu…

Let ${\mathbb F}_2$ be the second Hirzebruch surface, equipped with the family of Kähler–Einstein edge metrics $\omega_{\beta_1,\beta_2}$ described in the source, with…

Let $X$ be a Fano manifold. Let $N_k:=\dim H^0(X,-kK_X)$, and let the corresponding canonical point processes be defined by the Gibbs measures associated with determinants of bases…

Completeness conjecture. The metric $g_{KE}$ is complete near $p$; more precisely, there is an open neighborhood $V_{q,R}$ of $p$ such that for every $x\in V_{q,R}\setminus\{p\}$ a…

K-stability conjecture. Then $S_d$ is $K$-stable, and hence admits an orbifold Kähler–Einstein metric.

Let $S_d$ be a quasi-smooth and well-formed hypersurface of degree $d$ in $\mathbb{P}(a_0,a_1,a_2,a_3)$, where $a_0\leq a_1\leq a_2\leq a_3$ are positive integers. Define the index…

Let $\mathcal M_{s,p,n}$ be the manifold considered above, and let $s,p,n$ be its defining parameters. A line bundle $L$ on a projective variety $X$ is called big when its highest…

Kähler–Einstein relative singularity conjecture. If

Let $X$ be a smooth projective variety and let $D$ be a smooth divisor such that $(X,D)$ is asymptotically log Fano. For $\beta\in(0,1]$, write $(X,(1-\beta)D)$ for the correspondi…

Let $K\subset\mathbb{R}^n$ be a convex body with barycenter at the origin, and let $\Phi$ solve the Kähler–Einstein equation … For the simplex … the corresponding Hessian metric…