20 problems

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 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…

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

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 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…

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

Uniruledness conjecture. $X$ is uniruled, meaning covered by rational curves, 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 $X$ 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 simple compact Kähler manifold of dimension $n$, and write $q=h^1(X,\mathcal O_X)$ and $\kappa(X)$ for its irregularity and Kodaira dimension. A holomorphic $2$-form…

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…

Abelianity Conjecture. Every special compact Kähler manifold $X$ has an almost abelian fundamental group; that is, $\pi_1(X)$ has an abelian subgroup of finite index.

Algebraic-dimension conjecture. One has

Iitaka's conjecture. Then $X$ is a torus, up to finite étale cover.

Numerical inequality conjecture. If