24 problems

Let $X$ be a compact Kähler manifold and $D=\sum_{i=1}^{N}D_i$ be a simple-normal-crossing (snc) divisor on $X$. Let $F$ be a semi-positive line bundle on $X$, meaning that it admi…

Yau's conjecture. The manifold $X$ is projective and rationally connected.

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…

Conjecture. Theorem and Corollary remain true when $Y$ is a compact Kähler manifold of nonnegative Kodaira dimension.

Isotriviality conjecture for simple fibres. If $X_y$ is simple and $a(Y)=0$, then

Simple-manifold decomposition conjecture. $X_n$ is either bimeromorphically irreducible symplectic or Kummer. In particular, $X_n$ is Kummer if $n$ is odd.

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…

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

Kähler BDPP conjecture. The results of Boucksom, Demailly, Păun and Peternell concerning the pseudo-effectivity of the canonical bundle of the MRC quotient should remain valid for…

Let $X$ be a compact Kähler manifold of dimension $n$. Let $\mathrm{Psef}(X)$ be the pseudoeffective cone in $H^{1,1}(X,\mathbf{R})$, and let $\mathrm{Int}(\mathrm{Psef}(X)^\vee)$…

Let $X$ be a compact Kähler manifold and let $C\subset X$ be a smooth curve with ample normal bundle. Write $[C]$ for its cohomology class and let $\mathcal{K}(X)$ denote the Kähle…

Semisimplicity or virtual unipotence conjecture. The action $f^*_2$ is either semisimple or virtually unipotent. The mixed case is not possible for automorphisms of compact Kähler…

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…

Rational connectedness conjecture. $X$ is rationally connected, meaning that any two generic points are joined by some rational curve, if and only if

Pseudoeffectivity and uniruledness conjecture. The canonical class $K_X$ is pseudoeffective if and only if $X$ is not uniruled.

Fujiki-class finite-generation conjecture. The ring $R(X,\Delta)$ is a finitely generated $\mathbb C$-algebra. The source states that this conjecture is equivalent, after taking a…

Let $X$ be a normal complex analytic variety in Fujiki's class $\mathcal C$, let $\Delta$ be a $\mathbb Q$-divisor such that $K_X+\Delta$ is $\mathbb Q$-Cartier, and let $f:X\to Y$…

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…

Cone conjecture. Under this isomorphism, $S_{(\rho)}$ must be a cone in $C_{(0)}$.

Compactifiable universal cover conjecture. After passing to a finite étale cover, there exists a locally trivial fibration $X\rightarrow A$ with simply connected fibre $F$ onto a c…

Algebraic-dimension conjecture. One has

Numerical inequality conjecture. If