4 problems

Kummer conjecture. Every simple compact Kähler threefold is Kummer.

Let $Y$ be a smooth projective threefold of general type. The $m$-th pluricanonical map of $Y$ is the map defined by the $m$-th pluricanonical linear system. Demailly's conjecture.…

Fix a complex projective variety $X$, and let ${\cal G}_m$ be the set of nonsingular complex projective varieties $Y$ for which the $m$-th pluricanonical map…

Let $X$ be an algebraic variety over a field of characteristic zero. Let $R(X,Y)$ denote the set of dominant rational maps $f:X\rightarrow Y$, and let ${\cal F}={\cal F}(X)$ be the…