11 problems

Let $X$ be a nonsingular projective variety with $\kappa(X)=0$. Ueno's Conjecture K asserts that the Albanese map is birational to an étale fiber bundle over…

Let $X$ satisfy the assumptions that its cotangent bundle contains a nef locally free subsheaf $L^*\subset\Omega_X^1$ proportional to $K_X$. If $\kappa(X)=0$, then the product deco…

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

Schreieder–Kotschick conjecture. The following three statements are equivalent: (1) $X$ admits a global holomorphic 1-form without zeros; (2) $X$ admits a smooth real closed 1-form…

Let $X$ be an $n$-dimensional smooth complex quasi-projective variety with proper Albanese map … An abelian duality space of dimension $n$ is a space whose cohomology with coeffici…

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…

Noether–Castelnuovo conjecture. One should have

Let $M$ be a normal projective variety with at most canonical Gorenstein singularities. Let $\varphi\colon M\rightarrow C$ be a fibration onto a smooth curve $C$ such that…

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…

Manetti's equality-case conjecture. The equality

Albanese submersion conjecture. The Albanese map of $X$ is a surjective submersion.