16 problems

Let $f\colon(X,B)\rightarrow Z$ be a klt-trivial fibration, meaning that $(X,B)$ is Kawamata log terminal at the generic point of $Z$ and $K_X+B\sim_{\mathbb{Q}f}0$. The canonical…

Shokurov's conjecture. Fix a positive integer $r$ and a real number $0<\epsilon\leq 1$. There exists $\delta>0$ depending only on $r,\epsilon$ such that, if $\dim X-\dim Z=r$,…

Let $(X,B)$ be an lc pair such that $K_X+B$ is a $Q$-Cartier divisor, and let $f:(X,B)\to Z$ be an lc-trivial fibration. Let ${\mathbf D}_{\mo}$ denote its moduli b-divisor. Effect…

Positivity conjecture. Then $K_Y+B_Y$ is $\mathbb{Q}$-Cartier and the moduli part $M_Y$ is $\mathbb{Q}$-free: for some $\nu\in\mathbb{N}$, $\mathcal O_Y(\nu M_Y)$ is generated by g…

The Base Change Conjecture. Then $K_Y+B_Y$ is $\mathbb{Q}$-Cartier and, if $f':(X',B^{X'})\to Y'$ is induced by a birational base change $\sigma:Y'\to Y$, then

Let $X$ be a compact complex manifold with an lc-trivial contraction $\sigma\colon (X,R)\to Y$ over a smooth projective variety $Y$, where $R$ is a $\mathbb{Q}$-divisor on $X$. The…

Canonical bundle formula conjecture. There is

Generalized Kähler canonical bundle formula conjecture. If $f:(X,B+\boldsymbol\beta)\to Z$ is a generalized klt (respectively, lc) pair as above, then $(Z,B_Z+\boldsymbol\beta^Z)$…

Let $d$ be a positive integer and let $\Gamma\subset [0,1]$ be a DCC set of real numbers. A $\Gamma$-base-point-free $b$-divisor $\mathbf{D}$ on a normal projective variety $X$ is…

Let $d$ be a positive integer and let $\Gamma_0$ be a finite set of rational numbers. An lc-trivial fibration is a fibration $f:(X,B)\rightarrow Z$ with the usual lc-triviality con…

Finiteness conjecture. There is an open set $U\subseteq Y$ such that for every $y\in U$ the set

Let $(X,D)$ be an klt (respectively, lc) pair with dimension $d$, and let $f:(X,D)\to Z$ be an lc-trivial fibration. An $\mathbb R$-b-divisor ${\bf M}$ is $\Gamma$-base-point free…

Let $(X',B'+M')$ be a generalized sub-pair with data $X\to X'$ and $M$, where $B'$, $M'$ and $M$ are $\mathbb{Q}$-divisors. Let $f\colon X'\to Z'$ be a contraction such that … Assu…

Shokurov's conjecture. If $(X,\Delta)$ is $\epsilon$-log terminal, then there exists $\delta=\delta_{m,n}(\epsilon)>0$ and an effective moduli part $B$ such that $(Y,R+B)$ is…

Prokhorov–Shokurov conjectures. The following properties are conjectured:

Let $(X,\Delta)$ be a log klt pair and let $f\colon X\to Y$ be a surjective morphism such that the Kodaira dimension of $K_X+\Delta$ restricted to the general fibre is zero. Assume…