658 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…

Let $X$) be a compact Kähler manifold of dimension $n$, and let $[\omega_0]$ be any Kähler class on $X$. For a smooth $(1,1)$-form $T$, write $c_1(X)$ for the first Chern class of…

Let $(M,g,J,\omega)$ be a compact Einstein almost-Kähler manifold. Goldberg's conjecture. The almost-Kähler structure is Kähler; equivalently, the almost complex structure $J$ is i…

Abundance. The divisor $K_X+B$ is semiample in the following senses:

Let $(M,\omega)$ be a compact $n$-dimensional Kähler manifold with positive holomorphic bisectional curvature. Frankel's conjecture. Then $M$ is biholomorphic to $\mathbb{CP}^{n}$.…

Termination of flips. Any sequence of $K_X+B+\boldsymbol{\beta}_X$ flips is finite.

Tian's properness conjecture. A cscK metric exists if and only if the Mabuchi functional is coercive modulo $G$.

Let $(M^n,g)$ be a compact Kähler manifold of complex dimension $n$, let $comega_g$ be a Kähler metric, and let $comega_{cvarphi_s(t)}$ be a family of Kähler metrics in the Kähler…

Collins–Jacob–Yau conjecture. The dHYM equation admits a solution if and only if, for every proper subvariety $V\subset X$,

Let $X$ be a closed Kähler manifold of complex dimension $n$, with Kähler classes $alpha$ and $beta$. For a Kähler form $omega\in\beta$, a Kähler form $theta\in\alpha$ is a sub-sol…

Let $D\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. The Bergman metric of $D$ is the canonical Kähler metric associated with its Bergman kernel. A domain is biholomorphic…

Let $(X,L)$ be a polarised smooth projective variety over $\a9c$ of complex dimension $n$. An extremal metric is a Calabi extremal Kähler metric on $(X,L)$, and $(X,L)$ is relative…

Let $X$ be a compact Kähler manifold with infinite fundamental group, and let $\widetilde{X}$ be its universal cover. Carlson–Toledo conjecture. There exists a finite étale cover…

Ricci-flat tangent cone conjecture. If one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat. Consequently, the tangent cone at infinity should be unique; analogous qu…

Let $f:X\to S$ be a smooth family from a Kähler manifold $X$ onto a smooth, connected, relatively compact curve $S$. Assume that $(X,B+\boldsymbol{\beta})$ is a generalized klt pai…

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…

Streets–Tian conjecture. Any closed Hermitian-symplectic complex manifold $(X,J)$ admits a Kähler metric.

Peternell's conjecture. Any minimal Kähler variety has an algebraic approximation.

Yau's conjecture. Every compact Kähler manifold with positive holomorphic sectional curvature is projective and rationally connected.

SKT-to-Kähler hyperbolicity conjecture. If $X$ admits such a metric $\omega$, then $X$ is Kähler hyperbolic.

Let $X$ be a compact Kähler manifold with nef anticanonical bundle $-K_X$, and let $a:X\to A$ be its Albanese morphism to the Albanese variety $A$. Demailly–Peternell–Schneider con…

Let $(M,J)$ be a complex manifold. A $p$-Kähler structure on $(M,J)$ is a closed, pointwise transverse $(p,p)$-form. Alessandrini–Bassanelli conjecture. If $(M,J)$ admits a $p$-Käh…

Let $(X,\omega_0)$ be a compact Kähler manifold, and let…

Boucksom–Demailly–Păun–Peternell duality conjecture. The full cones $\mathcal{E}$ and $\mathcal{M}$ should be dual with respect to the Poincaré pairing for any compact Kähler manif…