19 problems

Chen–Nie conjecture. Let $(M^n,g)$ be a compact Hermitian manifold. Assume that the holomorphic section curvature of $D^r_s$ is a constant $c$. If $c\neq 0$, then $g$ must be Kähle…

Bismut space-form conjecture. If a compact Hermitian manifold $(M^n,g)$ has constant Bismut holomorphic sectional curvature $c$ and $c\neq 0$, then $g$ must be Kähler.

First-eigenvalue conjecture. One has

Chen–Zheng conjecture. If the holomorphic sectional curvature of $\nabla^b$ is a non-zero constant, then $g$ must be Kähler, hence a complex space form.

Hermitian space-form conjecture. If the Chern connection of $M$ has constant holomorphic sectional curvature, then $M$ must be either Kähler, hence a complex space form, or Chern f…

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$. For a real number $t$, let $\nabla^{(t)}=(1-t)\nabla+t\nabla^b$ be the $t$-Gauduchon connection, let…

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$, and let $H^r$ denote the Riemannian holomorphic sectional curvature, with $R^r$ the curvature of the R…

Let $M$ be a compact complex manifold admitting a Kähler metric $\omega$ with positive holomorphic sectional curvature. The holomorphic sectional curvature along a nonzero…

Let $M$ be a Stein manifold of complex dimension at least two, and let $A^2(M)$ be its Bergman space. Assume that $A^2(M)$ is base-point free and separates holomorphic directions.…

Let $M$ be a Stein manifold. Let $A^2(M)$ be its Bergman space, assumed to be base-point free, meaning that its sections do not vanish simultaneously at any point, and to separate…

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 $X$ be a compact Kähler manifold with semi-positive holomorphic sectional curvature. Structure conjecture. There should exist a smooth locally trivial morphism $X\to Y$ whose f…

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

Structure conjecture. There exists a smooth morphism $X\to Y$ such that a fiber is rationally connected and $Y$ admits a finite étale cover $A\to Y$ by an abelian variety $A$. More…

Let $X$ be a compact complex manifold of complex dimension $n>2$. RC-positivity conjecture. If $X$ has a Hermitian metric with positive holomorphic sectional curvature, then…

The optimal rank-form extension of Yau's conjecture. One should have

Ampleness conjectures.

Let $(X,\omega)$ be a compact Kähler manifold with strictly negative holomorphic sectional curvature, and let $K_X$ denote its canonical line bundle. Yau's ampleness conjecture. Th…

Let $M$ be a projective manifold with a Kähler metric of semi-negative holomorphic sectional curvature. Define $r_M$ as the invariant measuring the largest codimension of a maximal…