17 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 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…

Let $X$ be a projective manifold satisfying the assumptions that its cotangent bundle contains a nef locally free subsheaf $L^*\subset\Omega_X^1$ proportional to $K_X$. If…

Let $X$ be a smooth projective variety of dimension $d$ with generically finite Albanese map $a:X\rightarrow A$, let $P$ be a Poincaré bundle on $X\times\widehat A$, and let…

Let $X$ be a special manifold of dimension $n\geq 2$. A manifold is strongly irreducible special when every covering family with special general member is trivial in the sense defi…

Let $X$ be a smooth Kähler manifold with nef anticanonical divisor $-K_X$, and let … be its Albanese map. DPS conjecture. The Albanese map $\alpha_X$ is surjective. This conjecture…

Campana's MRC quotient conjecture. The MRC quotient is a projective morphism whose base is Calabi–Yau.

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

Conjecture B. The Albanese map is a locally isotrivial fibration onto the Albanese torus with rationally 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…

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…

Manetti's equality-case conjecture. The equality

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