On Morrison's cone conjecture for klt sufaces with $K_X \simeq 0$

Fano manifolds of index n-2 and the cone conjecture

The Morrison--Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair have rational, polyhedral fundamental domains. In a previous paper, we proved the conjecture for certain blowups of Fano manifolds of index n-1. In this paper, we consider the Morrison--Kawamata conjecture for blowups of Fano manifolds of index n-2.

Fano manifolds of index n-1 and the cone conjecture

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 $(X, Δ)$ have finite, rational polyhedral fundamental domains. Let $Z$ be an $n$-dimensional Fano manifold of index $n-1$ such that $-K_Z = (n-1) H$ for an ample divisor $H$. Let $Γ$ be the base locus of a general $(n-1)$-dimensional linear system $V \subset |H|$. In this paper, we verify the Morrison-Kawamata cone conjecture for the blow-up of $Z$ along $Γ$.

The cone conjecture for some rational elliptic threefolds

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 klt Calabi--Yau pairs in dimension 3 with non-zero boundary divisor.

关于维数至多为4的某些对的锥猜想

On the cone conjecture for certain pairs of dimension at most 4

本文利用极小模型纲领和反正则纤维化，在维数至多为4且Iitaka维数满足特定下界条件下证明了klt Calabi-Yau对的Morrison-Kawamata锥猜想。

Birational automorphism groups of projective varieties of Picard number two

We slightly extend a result of Oguiso on birational or automorphism groups (resp. of Lazić - Peternell on Morrison-Kawamata cone conjecture) from Calabi-Yau manifolds of Picard number two to arbitrary singular varieties X (resp. to klt Calabi-Yau pairs in broad sense) of Picard number two. When X has only klt singularities and is not a complex torus, we show that either Aut(X) is almost cyclic, or it has only finitely many connected components.

Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves

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 generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the Morrison-Kawamata cone conjecture from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in dimension 2 in the case of minimal rational elliptic surfaces.