11 problems

For every polarized smooth projective complex manifold $(X,L)$, the polarized manifold $(X,L)$ is $K$-polystable if and only if the Kähler class $c_1(L)$ admits a constant scalar c…

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

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…

Constant holomorphic sectional curvature conjecture. A compact Hermitian manifold with constant holomorphic sectional curvature is Kähler when the constant is non-zero and Chern fl…

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.

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…

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