35 research outputs found
Equidistribution des sous-variétés de petite hauteur
International audienceIn this paper, the equidistribution theorem of Szpiro-Ullmo-Zhang about sequences of small points in an abelian variety is extended to the case of sequences of higher dimensional subvarieties. A quantitative version of this result is also given
On the canonical degrees of curves in varieties of general type
A widely believed conjecture predicts that curves of bounded geometric genus
lying on a variety of general type form a bounded family. One may even ask
whether the canonical degree of a curve in a variety of general type is
bounded from above by some expression , where and are
positive constants, with the possible exceptions corresponding to curves lying
in a strict closed subset (depending on and ). A theorem of Miyaoka
proves this for smooth curves in minimal surfaces, with . A conjecture
of Vojta claims in essence that any constant is possible provided one
restricts oneself to curves of bounded gonality.
We show by explicit examples coming from the theory of Shimura varieties that
in general, the constant has to be at least equal to the dimension of the
ambient variety.
We also prove the desired inequality in the case of compact Shimura
varieties.Comment: 10 pages, to appear in Geometric and Functional Analysi
Hyperbolicity of varieties of log general type
These notes provide an overview of various notions of hyperbolicity for
varieties of log general type from the viewpoint of both arithmetic and
birational geometry. The main results are based on our paper entitled
"Hyperbolicity and uniformity of varieties of log general type." They are
expanded notes from a minicourse the authors gave as part of the Geometry and
arithmetic of orbifolds workshop at UQ\'AM.Comment: Addressed some inaccuracies and typos pointed out by the referees and
some readers. Slight change of title. To appear in CRM short courses
(Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces:
Hyperbolicity in Montr\'eal