5 problems

RC-positivity conjecture. There is no non-constant holomorphic map from $M$ to $N$.

Let $E$ be an ample vector bundle over a projective manifold $X$, and let $\operatorname{rank}(E)$ denote its rank. RC-positivity conjecture. There exists a smooth Hermitian metric…

Let $X$ be a projective manifold. Mumford's equivalent-conditions conjecture. If … for every $m\geq 1$, then each of the following holds: $X$ is uniruled; $K_X$ is not pseudo-effec…

Let $X$ be a projective manifold, let $K_X$ be its canonical bundle, let $K_X^{-1}$ be its anticanonical bundle, and let $\kappa(X)$ denote its Kodaira dimension. Uniruledness conj…

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…