284 problems

Let $V$ be the compact complex manifold underlying the forms in the statement, let $R$ be a compact Riemann surface with boundary, and let $\rho:V\times\partial R\to\mathbf{R}$ sat…

Let $\omega$ be the metric on $\mathbb C^k$ determined by Lemma, and let $\omega_p=iD_0\partial\bar{\partial}s^p$, where $s=|z|^2$ on $\mathbb C^k$. Say that a metric is asymptotic…

Rigidity conjecture. Either $T=0$ or $S=0$.

High-temperature uniqueness conjecture. For $\lambda\leq 0$, $\mu^\lambda$-cscK metrics are unique modulo the automorphism group.

Let $X$ be a compact Kähler Calabi-Yau manifold, and let $\alpha\in H^{1,1}_{\mathbb{R}}(X)$ be a class that is nef and big but not Kähler. A smooth $(1,1)$-form representing…

Let $M$ be an irreducible compact Kähler manifold with positive orthogonal bisectional curvature. Projective-space conjecture. Is $M$ necessarily biholomorphic to…

Let $G$ be a complex simple Lie group and let $P\subset G$ be a maximal parabolic subgroup. Assume that $G/P$ is not a compact Hermitian symmetric space, or, if $G/P$ is a compact…

Let $M$ be a nef Kähler manifold of dimension $n$. Suppose there is an epimorphism … where $\Gamma$ is a torsion-free nilpotent group of rank at least $2n$. Nilpotent quotient conj…

Let $(X,L)$ be a polarised variety, let $T$ be a maximal torus of automorphisms of $X$, and let $F_\chi(\alpha)$ denote the modified Futaki invariant associated with the extremal v…

Let the Kähler cone be the set of Kähler classes, and let the extremal Kähler cone be the subset consisting of classes represented by extremal Kähler metrics. Suppose the flow equa…

Let $(M,J)$ be a complex surface of positive first Chern class polarized by a Kähler class $\Omega$. The positive first Chern class convergence conjecture. There exists an initial…

Ample-normal-bundle conjecture. The algebraic dimension of $X$ should be at least one greater than the dimension of any positive-dimensional compact submanifold with ample normal b…

Let $V$ be a Kähler manifold, let $\mathcal{H}$ be the space of Kähler potentials, and let $\mathcal{H}_0$ denote its normalized subspace. For a normalized $\phi\in\mathcal{H}_0$,…

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 $(X,L)$ be a polarized manifold. Let $\mathcal{H}$ be the space of Kähler metrics in the class $c_1(L)$, let $\mathcal{H}^{\operatorname{NA}}$ be the space of smooth non-Archim…

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

Existence of log terminal models. The pair $(X,B+\boldsymbol{\beta})$ has a log terminal model.

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

Collins–Jacob–Yau hypercritical-phase conjecture. The set $\mathcal{H}_{\Omega}$ is non-empty and $\Omega$ has hypercritical phase if and only if, for every irreducible subvariety…

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 $\epsilon\in(0,a]$, and consider $\epsilon$-acK manifolds with parameters $(n,d,k,i_0,a)$. Harmonic-radius conjecture. The $W^{2,p}$ harmonic radius is uniformly bounded below…

Let $X$ be a $W^{2,p}\times W^{1,p}$ formally Kähler manifold with complex structure $J$ and metric $g$. Low-regularity Newlander–Nirenberg conjecture. The Newlander–Nirenberg theo…

Let $p\gg 0$, and let $X$ be a $W^{2,p}\times W^{1,p}$ formally Kähler manifold with almost-complex structure $J$ and nearby form $\omega_J$. Smooth Kähler structure conjecture.…

Generalized LeBrun equality. The equality above should hold for any compact complex $n$-manifold, at least when $K_M$ is big.

Interpolation conjecture. For each $s\in[0,1]$, the ray $\{u_t^s\}$ is also approximable.