32 problems

Strong rational connectedness conjecture. The smooth locus $X^{sm}$ of a log Del Pezzo surface $(X,\Delta)$ is strongly rationally connected.

Let $X^\partial$ be a smooth projective effective orbifold with $\kappa(X^\partial)=-\infty$. An orbifold Mori fibration is a fibration whose general fibers are orbifold rationally…

Rational connectedness criterion. $X$ is rationally connected if and only if

Let $X$ be a compact Kähler manifold, and let $TX$ denote its holomorphic tangent bundle. A Hermitian metric $c9$ on $TX$ is assumed to be uniformly RC-quasi-positive, or equivalen…

Let $(X,D_{\mathbfsymbol{m}})$ be a Campana pair, where $D_{\mathbfsymbol{m}}$ is its boundary divisor, and assume that the log-anticanonical divisor $-K_X-D_{\mathbfsymbol{m}}$ is…

Let $X$ be a compact Kähler manifold, let $T_X$ denote its holomorphic tangent bundle, and let BC-1 positivity refer to the paper's curvature-positivity notion for $T_X$. Yang's co…

Let $M$ be a Hermitian manifold with a Hermitian metric $g$. Assume that $g$ has positive or quasi-positive real bisectional curvature. Yang–Zheng's conjecture. Then $M$ is project…

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

Structural conjecture. There exists a fibration $\varphi:X\to Y$ such that $\varphi$ is locally constant, $Y$ is a compact Kähler manifold with $c_1(Y)=0$, and $F$ is rationally co…

Let $X$ be a compact Kähler manifold with nef anti-canonical bundle $-K_X$. A fibration is a surjective holomorphic map with connected fibers, and it is locally constant when its f…

sRC quotient structure conjecture. Such an orbifold morphism with all the stated properties exists.

Let $X$ be a rationally connected variety of dimension $n$, regularity $r$, and coregularity $c$. Assume that $G:=\pi_1^{\rm reg}(X)$ is finite. Coregularity structure conjecture.…

Let $X$ be a rationally connected variety, and let $(X,\Delta)$ be a dlt pair. Write $\pi_1^{\rm reg}(X)$ for the regional fundamental group. Finiteness conjecture for regional fun…

Rational connectedness conjecture. If $-(K_X+\Delta)$ is strictly nef, then $X$ is rationally connected.

Let $(X,D)$ be a log smooth pair, meaning that $X$ is smooth and $D$ has simple normal-crossing support. Its logarithmic tangent bundle is denoted by $T_X(-\log D)$. An sRC-quotien…

Campana–Cao–Matsumura's conjecture. There exists an orbifold morphism $\rho:(X,D)\to(R,D_R)$ such that:

Let $X$ be a smooth $n$-dimensional projective variety with nef anti-canonical bundle, and let $\mathcal{F}$ be a foliation on $X$. For some ample divisors $H_1,\ldots,H_{n-1}$ on…

Let $V$) be a divisorially canonical Fano variety, and let $X$ be a rationally connected variety with $\dim X=\dim V$. A rational dominant map is a rational map…

Structure conjecture for sRC quotients. There exists an orbifold morphism

Conjecture on quasi-positive curvature. If $X$ has a Kähler metric with quasi-positive holomorphic sectional curvature, then $X$ is a projective and rationally connected manifold.

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…

RC-positive tangent bundle conjecture. If $X$ is rationally connected, then $T_X$ is RC-positive.

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

Negative second scalar curvature conjecture. The manifold $M$ cannot be 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…