11 problems

Let $X$ be a compact Kähler manifold with nef anticanonical bundle $-K_X$, and let $a:X\to A$ be its Albanese morphism to the Albanese variety $A$. Demailly–Peternell–Schneider con…

Let $X$ be a compact Kähler manifold with nef anticanonical bundle, and let $\operatorname{alb}_X:X\to \operatorname{Alb}(X)$ be its Albanese map. Albanese submersion and isotrivia…

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 with nef anticanonical bundle, and let $\widetilde{X}$ be its universal covering. Universal-cover splitting conjecture. The universal covering…

Let $(X,\omega_X)$ be a compact Kähler manifold with nef anticanonical bundle. Torus or projective-line-fibration conjecture. One has … If equality holds for some Kähler metric, th…

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…