22 problems

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 $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial. Write $\mathcal{A}^e(X/Z)$ for the effective nef cone and $\mathcal{M}^e(X/Z)$ for th…

Arithmetic cone conjecture. Suppose $(X,B)$ is klt. Then there is a rational polyhedral cone $\Pi\subseteq\textnormal{Nef}^e(X/Y)$ such that

Let $f:(X,\Delta)\to S$ be a klt $K$-trivial fiber space. Define … and … A subset is a weak fundamental domain for a group action if its translates cover the space and every transl…

Let $(X,\Delta)$ be a klt Calabi–Yau pair. Let $\mathcal{B}^e(X)$ denote the effective cone appearing in the source, and let $PsAut(X,\Delta)$ act on it by pullback. Effective cone…

Let $(X,\Delta) \to S$ be a klt Calabi–Yau fiber space. Let $\Gamma_B$ and $\Gamma_A$ be the images of the pseudo-automorphism group $\operatorname{PsAut}(X/S,\Delta)$ and the auto…

Boundedness conjecture. Let $Y$ be an irreducible holomorphic symplectic manifold birational to $X$. The set

Fix a complex structure $t$ on $\X_t$ and let $\Mov(\X_t)_+$ be the convex hull of the rational movable divisor classes, equivalently the union of the proper transforms of the nef…

Geometric cone conjecture. Suppose $(X,B)$ is klt and $X$ is $\mathbb{Q}$-factorial. Then the number of $\textnormal{Aut}(X/Y,B)$-equivalence classes of faces of…

Let $X$ be a terminal Calabi–Yau variety, meaning a normal and $\mathbf{Q}$-factorial complex projective variety with $K_X\equiv 0$ and $h^1(X,\mathscr{O}_X)=0$. Write…

Type (1) face orbit conjecture. The pseudoautomorphism group $\operatorname{PsAut}(Y)$ acts with finitely many orbits on the Type (1) faces of $\operatorname{Mov}^{e}(Y)$ for every…

Totaro's cone conjecture. There exists a rational polyhedral cone $\Pi$ that is a fundamental domain for the action of $\operatorname{Aut}(X)$ on $\operatorname{Nef^e}(X)$, namely

Let $X$ be a Calabi–Yau variety, and consider its movable cone and the minimal models birational to $X$. Movable-cone Cone Conjecture. The variety $X$ should have only finitely man…

Let $(X,\Delta)$ be a klt Calabi–Yau pair. Let … where $N^1(X)_{\mathbb{Q}}$ is the rational Néron–Severi space of $X$, and let $\operatorname{Nef}^e(X)$ denote the nef effective c…

Kawamata–Morrison–Totaro Cone Conjecture. There exists a rational polyhedral cone $\Pi$ in $\operatorname{Nef}^e(X)$ that is a fundamental domain for the action of…

Let $X$ be a Calabi–Yau variety, let $D$ be a ${\mathbb Q}$-divisor in the closed movable cone $\operatorname{\overline{Mov}}(X)$, and consider the strict transform of $D$ under a…

Finiteness conjecture. The group $\Gamma_W$ is always finite. The preceding lemma proves finiteness under the stronger assumption that $X$ has terminal singularities; the conjectur…

Let $f:(X,\Delta)\to S$ be a klt Calabi–Yau fiber space. Let $\operatorname{Eff}(X/S)$, $\operatorname{Mov}(X/S)$, and $\operatorname{Amp}(X/S)$ denote the effective, movable, and…

Kawamata–Morrison Cone Conjecture. There exists a rational polyhedral cone $\Pi$ contained in the cone spanned by Chern classes of nef effective Cartier divisors,…

Kawamata's cone conjecture. There exists a fundamental domain for the action of $\operatorname{Aut}(X)$ on $\operatorname{Nef}(X)^e$.

The cone conjecture. The pair $(X,\Delta)$ has only finitely many small $\mathbb{Q}$-factorial modifications $X\dashrightarrow X_1$ up to pseudo-automorphisms of $X$.

The cone of curves of a variety with semi-ample anticanonical bundle is expected to be controlled by a group, which may be infinite. More specifically, the conjecture concerns a gr…