57 problems

Mumford's conjecture. The manifold $X$ 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…

Yau's conjecture. Every compact Kähler manifold with positive holomorphic sectional curvature is projective and rationally connected.

Yau's conjecture. The manifold $X$ is projective and rationally connected.

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…

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

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

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…

Miyanishi's conjecture. The smooth part $T-\operatorname{Sing} T$ is rationally connected.

A variety is unirational if it admits a dominant rational map from some projective space, and rationally connected if two general points can be joined by a rational curve. Uniratio…

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…

Let $V$ be a Fano variety with log terminal singularities. A variety is rationally connected when any two general points are connected by an irreducible rational curve. Zhang's con…

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

Folklore rational connectedness conjecture. Every rationally connected projective manifold carries a nonzero genus-zero Gromov–Witten invariant with two point insertions.

Sharpness conjecture. For every integer $n\geqslant 3$, there exists a field $L$ with $\operatorname{char}(L)\neq 2$ and a quadratic form $\varphi\in\mathsf{I}^nL$ with

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…

Campana's MRC quotient conjecture. The MRC quotient is a projective morphism whose base is Calabi–Yau.

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 $M$ be a projective manifold. It is rationally connected if a compact family of rational curves contains, for every pair of points $x,y\in M$, a curve passing through both poin…