Related papers: The Kawamata-Morrison Cone Conjecture for Generali…
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 prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.
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…
The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture…
Enriques manifolds are non--simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of this note is to prove the Morrison--Kawamata…
The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using…
The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M.…
Let $A$ be an abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and an effective horizontal divisor $\mathcal{D} \subset \mathcal{A}$. We study $(S,…
A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both…
In an earlier paper the author proved one divisibility of Perrin- Riou's Iwasawa main conjecture for Heegner points on elliptic curves. In the present paper, that result is generalized to abelian varieties of GL2-type (i.e. abelian…
We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. This result corresponds through the Green-Griffiths-Lang's conjecture to a…
Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence…
This article contributes to the study of the generic part of the cohomology of Shimura varieties. Under a mild restriction of the characteristic of the coefficient field, we prove a torsion vanishing result for Shimura varieties of abelian…
The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semiabelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite…
This is a survey of the Kawamata-Morrison cone conjecture on the structure of Calabi-Yau varieties and more generally Calabi-Yau pairs. We discuss the proof of the cone conjecture for algebraic surfaces, with plenty of examples. We show…
In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian…
Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-G\"ottsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an…
A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected-Hodge-type; such schemes should include all…
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…
In this paper, we prove the Geometric Arveson-Douglas Conjecture for a special case which allow some singularity on $\partial{\mathbb{B}_n}$. More precisely, we show that if a variety can be decomposed into two varieties, each having nice…