Related papers: On the cone conjecture for certain pairs of dimens…

We verify the Morrison--Kawamata conjecture for a certain class of rational threefolds, namely blowups of P^3 in the base locus of a net of quadrics with no reducible members. This seems to be the first verified case of the conjecture for…

We prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs in dimension 2. That is, for a large class of rational surfaces as well as K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the…

We formulate an effective cone conjecture for klt Calabi--Yau pairs $(X,\Delta)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms…

Assuming the Morrison-Kawamata cone conjecture for the generic fiber of a Calabi-Yau fibration and the abundance conjecture, we show (1) the finiteness of minimal models, (2) the existence of a weak rational polyhedral fundamental domain…

We relate the Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces to the existence of Shokurov polytopes. For K3 fibrations, the existence of (weak) fundamental domains for movable cones is established. The relationship between…

We prove some finiteness results on the movable cone for mildly singular 3-folds with semiample anticanonical bundle, giving some evidence for the Morrison--Kawamata cone conjecture for klt pairs.

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence,…

We prove some version of Morrison's conjecture on the cone of divisors for Calabi-Yau fiber spaces with non-trivial base pace whose total space is 3-dimensional.

Let $X\rightarrow S$ be a fibration of relative dimension at most two and let $(X,\Delta)$ be a klt pair for which $K_X+\Delta \equiv_S 0$. We show that there are only finitely many Mori chambers and Mori faces in the movable effective cone…

We prove the Morrison--Kawamata cone conjecture for projective primitive symplectic varieties with $\Q$-factorial and terminal singularities with $b_2\geq 5$, from which we derive for instance the finiteness of minimal models of such…

Following a recent work of Oguiso, we calculate explicitly the groups of automorphisms and birational automorphisms on a Calabi-Yau manifold with Picard number two. When the group of birational automorphisms is infinite, we prove that the…

Let $X$ be a Calabi--Yau variety of Picard number two with infinite birational automorphism group. We show that the numerical dimension $\kappa^{\mathbb{R}}_{\sigma}$ of the extremal rays of the closed movable cone of $X$ is $\dim X/2$.…

We prove that for a fixed klt Calabi-Yau pair $(X,\Delta)$ of dimension $3$, the set of fiber space structures of $X$ is finite up to $Aut(X,\Delta)$.

Given a fibration $f: X \to Y$ with normal general fibre $X_y$, over a field of any characteristic, we establish the Iitaka-type inequality $\kappa(X,-K_X) \leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series…

We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $\kappa\ge 2$.

We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite…

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is…

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, \Delta)$ have finite, rational…

For a Calabi-Yau manifold $X$, the Kawamata - Morrison movable cone conjecture connects the convex geometry of the movable cone $\overline{\mathrm{Mov}}(X)$ to the birational automorphism group. Using the theory of Coxeter groups, Cantat…

We show that for any smooth Calabi-Yau threefold $X$ of Picard number $2$ with infinite birational automorphism group, the numerical dimension $\kappa_\sigma$ of the extremal rays of the movable cone of $X$ is $\frac{3}{2}$. Furthermore, we…