43 problems

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…

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.

A compact Kähler manifold is a compact complex manifold admitting a Kähler metric. Its holomorphic sectional curvature is the sectional curvature of the metric restricted to each c…

Let $(M^n,g)$ be a compact Hermitian manifold with $n\geq 2$. Let $H^s$ denote the holomorphic sectional curvature of the Strominger connection. Strominger curvature conjecture. If…

Yau's conjecture. If $X$ has positive holomorphic sectional curvature, then $X$ is projective and rationally connected.

Minimal-model conjecture. If $Y$ has at most terminal singularities and $K_Y$ is a nef $\mathbb{Q}$-Cartier divisor, then $Y$ is smooth and the rational map $f$ is a morphism. More…

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 $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. The holomorphic sectional curvature is $H(X)=R(X,\overline{X},X,\overline{X})/|X|^{4}$ for nonzero $(1,0)$-vectors…

First-eigenvalue conjecture. One has

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. Let $H$ denote its holomorphic sectional curvature and let $R$ denote its curvature tensor. The constant holomorphic…

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

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 $n\geq 2$ 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 $M$ 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…

A Stein manifold is a complex manifold admitting a proper holomorphic embedding into some complex Euclidean space. Its Bergman metric is the metric induced by its Bergman kernel wh…

Constant holomorphic sectional curvature conjecture. The metric is Chern-flat if $c=0$ and Kähler if $c\neq 0$.

Let $(M^n,g)$ be a compact Hermitian manifold with $n\geq 2$. Denote by $R$ and $R^c$ the curvature tensors of the Levi-Civita and Chern connections, and by $H$ and $H^c$ their hol…

Yau's ampleness conjecture. The canonical line bundle $K_X$ is ample.

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…