829 problems

Let $X$ be a projective variety over $\mathbb C$. Define the analytic special set $\operatorname{Sp}_{\mathrm{an}}(X)$ as the Zariski closure in $X$ of the union of the images of a…

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 $X$ be a compact complex manifold. A balanced metric is a Hermitian metric whose fundamental form $omega$ satisfies $domega^{n-1}=0$, and a pluriclosed metric is a Hermitian me…

Let $X$ be a smooth complex projective variety. Its fundamental group is large if, for every closed irreducible positive-dimensional subvariety $Z\subset X$, the image … is infinit…

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

Green–Griffiths conjecture. Every entire curve $f\colon\mathbb C\to X$ in a projective manifold $X$ of general type is algebraically degenerate.

Let $X$ be a minimal class VII surface with positive second Betti number $b_2(X)>0$. A global spherical shell is a smooth $3$-sphere with a neighborhood biholomorphic to a neighbor…

Let $E$ be an ample vector bundle over a compact complex manifold $M$. A Hermitian metric on $E$ is Griffiths-positive when its curvature is positive in the Griffiths sense. Griffi…

Let $(X^n,\omega_0)$ be a compact Kähler manifold with semiample canonical bundle $K_X$. Let $\omega(t)$ solve the normalized Kähler–Ricci flow … Let $f:X\to Y$ be the canonical-mo…

Mumford's conjecture. The manifold $X$ is rationally connected if and only if

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 $(\mathcal{X},\omega)$ be a closed Kähler manifold and let $\alpha$ be a closed nonnegative $(1,1)$-form satisfying … Let $(\omega(t),\alpha(t))$ solve the Kähler–Ricci flow co…

Popovici's conjecture. Every compact ∂∂̄-manifold is balanced.

Let $\Omega \subset \mathbb{C}^n$ be a smoothly bounded strictly pseudoconvex domain. Its Bergman kernel has an expansion … where $\psi$ is a defining function for $\Omega$, meanin…

Let $\mathcal F_{\bm A}$ be the phase-retrieval map associated with a matrix $\bm A\in\mathbb C^{M\times N}$, and let $N_{\min}(\mathbb C^M)$ denote the smallest $N$ for which some…

Let $(X^n,\omega)$ be a compact Hermitian manifold and let $\alpha$ be a closed real $(1,1)$-form whose Bott–Chern class $[\alpha]$ is nef. A class is big if it contains a current…

Let $L$ be a holomorphic line bundle on a complex manifold. Suppose that $L$ is semi-positive and positive at least at one point. Grauert–Riemenschneider conjecture. Then $L$ is bi…

Let $X$ be a compact Kähler manifold, and let $TX$ denote its holomorphic tangent bundle. A Hermitian metric $c9$ on $TX$ is assumed to be uniformly RC-quasi-positive, or equivalen…

Let $X$ be an $n$-dimensional compact complex manifold, and let $\alpha$ be a real $d$-closed $C^\infty$ $(1,1)$-form on $X$. Define … The BottChern cohomology class…

Feldman–Ilmanen–Knopf conjecture. The set $\Sigma$ should be an analytic subvariety of $X$. This predicts that the locus where a finite-time Kähler–Ricci-flow singularity fails to…

Gauduchon's conjecture for balanced metrics. One can always find a balanced metric in the prescribed Bott–Chern cohomology class whose first Chern–Ricci form coincides with $\Psi$.

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

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

Bedford's conjecture. The basin $\Omega_{\{F_j\}}$ is biholomorphic to $\mathbb{C}^k$. While the conjecture remains open, it is known that $\Omega_{\{F_j\}}$ is an increasing union…

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…