109 problems

Let $(X,\Delta)$ be a Klt pair, and let $f:X\to Y$ be an algebraic fiber space, where $X$ and $Y$ are smooth projective varieties of dimensions $n$ and $m$, respectively. Let $F$ b…

Kollár's conjecture. If

Let $X$ be a smooth projective variety over $\mathbb{C}_p$, and let $X^{\mathrm{an}}$ denote its analytification. Kodaira-dimension conjecture. If there is a $p$-adic analytic map…

Hacon–McKernan conjecture. A phenomenon analogous to the existence of a uniform integer $r_n$ such that $|rK_X|$ is birational for all sufficiently large $r$ should hold for every…

Let $X$ be a smooth projective variety of Kodaira dimension zero. Kollár's conjecture. The fundamental group $pi_1(X)$ is virtually abelian. This predicts strong restrictions on th…

Let $X$ be a smooth projective variety over the complex numbers. Its holomorphic Kodaira dimension is denoted by $\kappa^h(X)$, and $X$ is uniruled if through a general point of…

Log Iitaka conjecture. One has

Lang's conjecture. $X$ is rationally swept out by abelian varieties.

Let $f:X\to Y$ be an algebraic fiber space of the type considered in the paper, let $(X,\Delta)$ be the associated pair, and let $m$ be the positive integer appearing in the relati…

Let $X$ be a smooth projective variety with $\kappa(X)=0$, and let $\alpha:X\rightarrow A$ be its Albanese map, with general fibre $F$. Ueno Conjecture K. The map $\alpha$ is surje…

Ueno's Conjecture K strengthening. There exists an isogeny $A' \to A$ such that

Let $f\colon X\to Y$ be an algebraic fiber space between smooth projective varieties, let $V\subseteq Y$ be the open subset over which $f$ is smooth, and let $F$ be its general fib…

Iitaka–Viehweg conjecture. The Kodaira dimension of $X$ satisfies

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 $X$ be a compact complex manifold. Its Kodaira dimension is denoted by $4\kappa(X)4$, and its numerical Kodaira dimension $4\nu(X)4$ is defined by requiring $4\nu(X)\geq k4$ wh…

Dimension-free extension conjecture. All the preceding results should hold without restriction on $\dim A$.

For a projective variety $Z$, let $\kappa(Z)$ denote its Kodaira dimension, and say that $Z$ is uniruled if through a general point of $Z$ there passes a rational curve. Uniruledne…

Let $X$ be a smooth projective variety. A variety is uniruled if there are a variety $B$ of dimension $\dim X-1$ and a dominant rational map…

Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $\kappa(\overline{\mathcal{M}}_g)$ denote its Kodaira dimension. Low-genus Kodaira-dimensi…

Let $f^\circ:X^\circ\to S^\circ$ be a smooth family of canonically polarized varieties. Write $\operatorname{Var}(f^\circ)$ for its variation and $\kappa(S^\circ)$ for the logarith…

Let $F$ be a projective manifold of dimension $d$ with Kodaira dimension $\kappa(F)=0$, and let $L$ be an almost strictly nef line bundle, meaning that $L$ is the pullback of a str…

Let $X$ be a smooth projective variety. A variety is uniruled if through a general point of it there passes a rational curve, and its Kodaira dimension is denoted by $\kappa(X)$. U…

Let $X$ be a projective manifold of dimension $n$ such that $\kappa(X)=0$ and $\chi(\mathcal O_X)\neq 0$. Almost-abelianity conjecture. The conclusion known in dimension $4$ should…

Let $X$ be a projective manifold. Suppose that, for some positive integer $N$, … where $A$ is an effective divisor and $B$ is a pseudo-effective line bundle. Campana's abundance-ty…

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