9 problems

Let $X$ be a compact Kähler manifold with nef anti-canonical bundle $-K_X$. A locally constant fibration is a fibration that is locally constant in the sense of Definition 2.3 of M…

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…

Albanese fibration conjecture. The Albanese morphism of $X$ is a locally trivial fibration with connected fibers.

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…

Let $X$ be a compact Kähler manifold with nef anticanonical class. Denote its universal cover by $\widetilde{X}$. Decomposition conjecture. The universal cover decomposes as a prod…

Let $X$ be a compact Kähler manifold such that $-K_X$ is nef, and let $p:X\rightarrow Y$ be its Albanese map. Demailly–Peternell–Schneider conjecture. The map $p$ is locally trivia…

Let $X$ be a compact Kähler manifold of dimension $n$ with nef anticanonical bundle, and let $\omega_1,\ldots,\omega_{n-2}$ be nef classes. Kawamata's second-Chern-class conjecture…

Let $X$ be a compact Kähler manifold of dimension $n$ with nef anticanonical bundle, and let $\omega_1,\ldots,\omega_{n-2}$ be nef classes. Kawamata's second-Chern-class conjecture…

Let $X$ be a projective manifold. The anticanonical bundle is nef when $-K_X$ has nonnegative degree on every curve. The Albanese map is the canonical morphism from $X$ to its Alba…