284 problems

Hodge-theoretic Hopf conjecture. For every $p$ with $0\leq p\leq n$,

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

Yau's uniformization conjecture. If $M$ has these properties, then $M$ is biholomorphic to the standard $m$-dimensional complex Euclidean space

Lejmi–Szekelyhidi's conjecture. Under this normalization, the $J$-equation admits a unique smooth solution $\omega$ if and only if, for every $m$-dimensional analytic subvariety…

Let $X$ be a smooth complex compact Kähler variety, and let $\widetilde{X}$ denote its universal cover. Shafarevich's conjecture. The universal cover $\widetilde{X}$ is holomorphic…

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. Let $\{\beta\}\in H^{1,1}(X,\mathbb{R})$ be a real $(1,1)$-class with smooth representative $\beta$. The…

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

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. If the Calabi flow exists for all time and there exists a cscK metric in the Kähler class, then the Calabi flow converges to a cscK metric.

Let $\varphi$ be a smooth initial Kähler potential, and let the twisted Calabi flow be the evolution equation introduced in the paper for the corresponding Kähler metrics and twist…

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

Let $X$ be a compact Kähler manifold. A class $\alpha \in H^{1,1}(X,\mathbb{R})$ is semi-ample if there exists a holomorphic contraction $\rho:X\to X'$ to a normal analytic space…

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…

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. For a $(1,0)$-vector $X$, let $\mathcal{C}_{\alpha,\beta}(X)$ denote the mixed curvature, formed from the relevant Ch…

Let $M$ be a compact Kähler manifold with Kähler metric $\omega$, and let $\alpha$ be a real $(1,1)$-form representing $c_{1}(M)$. Calabi's conjecture. There exists a unique Kähle…

Kotschick's conjecture. The following conditions are equivalent:

Wu–Zheng's conjecture. There exists a finite étale cover $X'$ of $X$ admitting a smooth fibration in complex tori

Let $(M,h)$ be a simply-connected complete Kähler manifold satisfying … for two positive constants $A$ and $B$. Let $\mathfrak{K}$ denote the Kobayashi metric on $M$. Greene–Wu's c…

Let $(X,B)$ be a klt pair, where $X$ is a normal $Q$-factorial compact Kähler variety. Let $\alpha\in H^{1,1}_{\rm BC}(X)$ be a nef class on $X$. If $\alpha-(K_X+B)$ is nef and big…

Calabi–Chen conjecture. Initiating from any smooth Kähler potential, the Calabi flow always exists globally.

Chen's properness conjecture. A constant scalar curvature Kähler metric exists if and only if the $K$-energy is proper with respect to geodesic distance.

Let $(M,[\omega],J)$ be a compact Kähler manifold, let … and let $\mathcal K$ be the K-energy on Kähler potentials. A $C^{1,\bar 1}$ potential is a weak Kähler potential whose comp…

Let $(X,\omega)$ be a compact Kähler manifold, let $(\mathcal E^1,d_1)$ be the metric completion of the space of Kähler potentials, and let the extended Mabuchi K-energy be a map…

Chen's conjecture. The Calabi flow exists for all time.

Let $(M,\omega,\chi,J)$ be the initial twisted Kähler data, and suppose that the twisted Calabi flows have global existence. The transformed data along the flow may be written…