Related papers: The Morrison Cone Conjecture under Deformation

We give a counterexample of Morrison's cone conjecture for a strict Calabi-Yau threefold.

We investigate the moduli theory of Calabi--Yau threefolds, and using Griffiths' work on the period map, we derive some finiteness results. In particular, we confirm a prediction of Morrison's Cone Conjecture.

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…

In this paper, a family of smooth multiply connected Calabi--Yau threefolds is investigated. The family presents a counterexample to global Torelli as conjectured by Aspinwall and Morrison.

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…

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.

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…

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 describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone…

We give a proof of the Morrison--Kawamata cone conjecture for abelian varieties. The proof is a straightforward deduction from well-known results on the real endomorphism algebra of an abelian variety and reduction theory for self-dual…

We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the…

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 introduce a property of convex cones, being "well-clipped", that is inspired by the work of several complex algebraic geometers on the Morrison-Kawamata cone conjecture. That property is satisfied by movable cones of divisors on various…

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their N\'eron--Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties,…

This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general…

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…

Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration $f : Y \rightarrow \mathbb{P}^1$ with $-K_Y = f^*\mathcal{O}(1)$. We show that the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces…

We prove the weak relative Kawamata-Morrison movable cone conjecture for K-trivial fibrations whose very general fibre is a quotient, by a finite group of automorphisms acting freely in codimension one, of a product of certain Calabi-Yau…

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 prove modularity for a huge class of rigid Calabi-Yau threefolds over $\Q$. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular.