603 problems

DK hypothesis. K-equivalence should imply an equivalence … More generally, if $p^*K_X\geq q^*K_Y$, one expects a fully faithful embedding from $D^b\operatorname{Coh}(Y)$ into…

Let $k$ be an algebraically closed field of characteristic $0$. Every projective klt Fano orbifold $(X,\Delta)$, meaning a klt orbifold pair with $-(K_X+\Delta)$ ample, is Campana…

For every integer $d\ge 1$ and every real number $\varepsilon>0$, the class of $d$-dimensional $\varepsilon$-log-canonical Fano varieties over a field, equivalently varieties $X$ w…

For a projective morphism $f\colon X\to S$ and a relatively pseudo-effective $\mathbb{R}$-Cartier divisor $D$ on $X$, the relative non-nef locus equals the relative restricted base…

For every smooth complex projective variety $X$ of dimension $n$ and every ample Cartier divisor $L$ on $X$, the adjoint divisor $K_X+(n+1)L$ is globally generated.

Given a proper morphism $f:\mathcal{X}\to\mathcal{Z}$ with connected fibers between smooth separated Deligne--Mumford stacks, and a smooth effective Cartier divisor…

The Generalised Cone Conjecture asserts that the cone-conjecture theorem for Calabi--Yau surfaces extends to a broader class of algebraic surfaces beyond the Calabi--Yau case. The…

Let $X$ be a nonsingular toric weak Fano $d$-fold. Two such varieties are weakly-F-equivalent if they are connected by a sequence of equivariant blow-ups, blow-downs, and flops thr…

Let $A={\mathbb{A}}_k^N$ be affine space over an algebraically closed field $k$, with origin $0\in A$, and let ${\frak{a}}^e$ be a multiideal whose exponent set is $e$. ACC conject…

Let $X$ be a smooth, projective variety over $\mathbb{C}$, and write $h^{p,0}(X)$ for the Hodge number measuring global holomorphic $p$-forms. Bloch's conjecture. If … for every…

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…

Let $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial and $\Delta$ is an $\mathbb{R}$-boundary. Let $\mathcal{A}^e(X/Z)$ and…

Let $X$ be a singular variety and let … be a resolution of rational singularities. Write $\textbf{D}^{b}(X)$ and $\textbf{D}^{b}(\widetilde{X})$ for their bounded derived categorie…

Let $f:X\to S$ be a $K$-trivial fiber space, meaning a normal $\mathbb Q$-factorial klt pair $(X,\Delta)$ endowed with a proper surjective morphism with connected fibres such that…

Minimal model conjecture. Then $(X/Z,B,\mathbf{M})$ has a minimal model.

BAB conjecture. The set of $\epsilon$-lc Fano varieties of dimension $d$ forms a bounded family.

Borisov–Alexeev–Borisov conjecture. There is a positive number $\delta(n)$ such that

Let $X$ be a quasismooth Fano threefold weighted complete intersection of codimension $2$ and index $1$, belonging to a deformation family indexed by $I$. Let $I_{br}$, $I_F$, and…

Birational invariance conjecture. The set-theoretic image $\operatorname{im}(h^r_X)$ of the Hitchin morphism is invariant under smooth birational modifications.

Lower semi-continuity conjecture. The function $x\mapsto \operatorname{mld}_x(X,\mathfrak{a})$ is lower semi-continuous.

Shokurov's toric-model conjecture. Every maximal log Calabi–Yau pair $(X,\Delta_X)$ whose underlying variety $X$ is a rational $3$-fold has a toric model.

Schwede–Smith conjecture. $X$ is of Fano type.

Existence of Complements Conjecture. There exists a positive integer $n$, depending only on $d$ and $\Gamma$, such that every such pair $(X/Z\ni z,B)$ has an $n$-complement…

Morrison's cone conjecture.

Effective Iitaka fibration conjecture. There exists a positive integer $m_d$ depending only on $d$, such that for every smooth projective variety $X$ of dimension $d$ with non-nega…