On Albanese and Picard 1-motives with $${\mathbb{G}_a}$$ -factors

We construct Laumon-1-motives Pic⁺_a(X), Alb^-_a(X), Pic^-_a(X){{\rm Pic}^+_a(X), {\rm Alb}^-_a(X), {\rm Pic}^-_a(X)}, and Alb⁺_a(X){{\rm Alb}^+_a(X)} associated to an algebraic variety X with complete singular locus, whose associated étale (Deligne-)1-motives coincide with Picard and Albanese motives constructed by Barbieri Viale and Srinivas.     

Crystalline realizations of 1-motives

We consider the crystalline realization of Delignes 1-motives in positive characteristics and prove a comparison theorem with the De Rham realization of (formal) liftings to zero characteristic. We then show that one dimensional crystalline cohomology of an algebraic variety, defined by forcing universal cohomological descent via de Jongs alterations, coincides with the crystalline realization of the Picard 1-motive, over perfect fields of characteristic >2.     

Universal extension crystals of 1-motives and applications

We use the crystalline nature of the universal extension of a 1-motive M to define a canonical Gauss-Manin connection on the de Rham realization of M. As an application we provide a construction of the so-called Manin map from a motivic point of view.     

1-motivic sheaves and the Albanese functor

Using sheaf theoretic methods, we define functors and . The functor extends the one in [L. Barbieri-Viale, B. Kahn, On the derived category of 1-motives, I. Prépublication Mathématique de l’IHÉS (M/07/22), June 2007, 144 pages] to non-necessarily geometric motives. These functors are then used to define higher Néron-Severi groups and higher Albanese sheaves.     

The Fontaine–Ogus realization of Laumon 1-motives

Abstract

We construct the (filtered) Ogus realization of Laumon 1-motives over a number field. This realization extends the functor defined on Deligne 1-motives by Andreatta, Barbieri-Viale and Bertapelle.     

On semistability of Albanese maps

In this note, we show that the Albanese map for a smooth projective variety X with numerically effective anticanonical bundle is surjective, equi-dimensional and semistable.     

Comparison of Picard groups in dimension 1

We compare two Picard groups in dimension 1. Our proofs are constructive and the results generalize a theorem of J. Sands [11]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Albanese torsors and the elementary obstruction

We show that the elementary obstruction to the existence of 0-cycles of degree 1 on an arbitrary variety X, over an arbitrary field, can be expressed in terms of the Albanese 1-motives associated with dense open subsets of X. Arithmetic applications are given.     

The l-component of the unipotent Albanese map

We prove a finiteness theorem for the local l ≠ p-component of the -unipotent Albanese map for curves. As an application, we refine the non-abelian Selmer varieties arising in the study of global points and deduce thereby a new proof of Siegel’s theorem for affine curves over of genus one with Mordell–Weil rank at most one.     

On generalizations of the Albanese variety

In this note the relation between two generalizations of the classical Albanese variety, namely the one for smooth non-complete varieties by Faltings and Wüstholz and the one for singular varieties by Esnault, Srinivas and Viehweg, is investigated over the field of complex numbers. This work was supported by a grant of the Deutsche Forschungsgemeinschaft.     

On varieties of maximal Albanese dimension

We study the birational geometry of varieties of maximal Albanese dimension in this article. Given a non-birational, generically finite, and surjective morphism f: X → Y between varieties of maximal Albanese dimension, we show that the plurigenera P _m (X) and P _m (Y) for some m?≥ 2 could be equal only in very restrictive situations. We also prove that the 5-th pluricanonical map of a variety of maximal Albanese dimension always induces the Iitaka model.     

Ulrich bundles on quartic surfaces with Picard number 1

In this note, we prove that there exist stable Ulrich bundles of every even rank on a smooth quartic surface $X?{\mathbb{P}}^3$ with Picard number 1.     

Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces

In this paper we study nonlocal Cauchy problems for differential equations in Banach spaces. Using Picard and weakly Picard operators technique and suitable Bielecki norms, some existence, uniqueness and data dependence results are obtained under some mild conditions. As an application, we also discuss a class of impulsive Cauchy problems by adapting the same methods.     

On some smooth projective two-orbit varieties with Picard number 1

We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbit varieties with Picard number 1 that satisfy this latter property.     

Pluricanonical maps of varieties of maximal Albanese dimension

Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of in terms of the set . An immediate consequence of this is that the Kodaira dimension is invariant under smooth deformations. We then study the pluricanonical maps . We prove that if X is of general type, is generically finite for and birational for . More generally, we show that for the image of is of dimension equal to and for , is the stable canonical map. Received July 7, 2000 / Published online April 12, 2001