23 problems

On a hyperhermitian manifold $(M,I,J,K,g)$, let $4\Omega=\omega_J-i\omega_K4$, where $4\omega_J4$ and $4\omega_K4$ are the fundamental forms associated with $J$ and $K$, and let…

Let $M$ be a compact Hermitian manifold with Hermitian metric $g$, and let its Chern or Levi-Civita connection have holomorphic sectional curvature equal to a constant $c$. Constan…

Let $M$ be a non-compact Kähler manifold with positive bisectional curvature. Yau's uniformization conjecture. The manifold $M$ is biholomorphic to the complex Euclidean space. Thi…

Let $M^n$ be a compact $n$-dimensional Kähler manifold with positive first Chern class $C_1(M)$, and consider the normalized Ricci flow … A gradient Kähler–Ricci soliton is a solut…

Donaldson's conjecture. The J-equation is solvable under the assumption

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

Yau–Yang curvature integral conjecture. There exists a constant $C>0$ such that, for every $r>0$ and every integer $k$ with $1\leq k\leq n$,

Z-critical analogy conjecture. The Z-critical equation is the correct analogue of Bridgeland stability on the PDE side.

Let $n\geq 2$ be an integer. A compact Kähler manifold is a compact Kähler manifold with its holomorphic sectional curvature nonnegative, and an ample anti-canonical line bundle me…

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 $(M^n,g)$ be a compact Hermitian manifold with constant real bisectional curvature, meaning that $B_g(e,a)=c|a|^2$ for every point $p\in M$, every unitary tangent frame $e$, an…

Let $(M^n,g)$ be a compact Hermitian manifold, and let $H$ denote its holomorphic sectional curvature. Suppose that $H$ is equal to a constant $c$. Constant holomorphic sectional c…

Let $S^2$ be the Riemann sphere, let $p_1,\ldots,p_d$ be distinct points, and let … be a non-planar minimal surface with embedded planar ends. Evenness conjecture. Then $d\geq 4$ i…

RC-positivity equivalence conjecture. The following statements are equivalent:

Quasi-positive curvature conjecture. Then $X$ is projective and rationally connected.

RC-positivity equivalence conjecture. The vector bundle $\mathscr E$ is (uniformly) RC-positive if and only if ${\mathcal O}_{\mathscr E^*}(-1)$ is uniformly RC-positive.

Infinite-time existence conjecture. Under these assumptions, the Kähler–Ricci flow has a solution for infinite time. This conjecture extends the cited existence result under the ad…

Let $\varphi$ be a solution of the Fu–Yau equation in Theorem. Uniqueness conjecture. The solution $\varphi$ is unique. The conjecture asserts that the condition in the uniqueness…

Vanishing real bisectional curvature conjecture. If the real bisectional curvature is constantly equal to $c$, then $c=0$ and

Let $M^n$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature. Suppose that $M$ admits a nonconstant holomorphic function with polynomial growth and tha…

Let $M=E\times S$ and let $\omega(t)$ solve the normalized Kähler–Ricci flow on $M$ with initial Kähler metric $\omega_0$. Theorem (i) states that for any $\beta\in(0,1)$,…

Let $X=M\times E$, where $M$ and $E$ are compact Kähler manifolds, and let $\omega(t)$ solve the normalized Kähler–Ricci flow … In the case $m=n=1$, where the convergence of the ba…

Uniform lower bound conjecture. There is a constant $C$ such that