Albanese and Picard 1-motives

TL;DR

This paper develops algebraic cohomological and homological 1-motives for algebraic varieties in characteristic zero, generalizing classical concepts and proving conjectures related to their realizations and functorial properties.

Contribution

It introduces algebraically defined 1-motives for any algebraic variety, extending classical Albanese and Picard varieties, and proves Deligne's conjecture for their realizations.

Findings

Proves Deligne's conjecture for mixed Hodge structures.

Constructs universal Abel-Jacobi maps for zero cycles.

Establishes functoriality and invariance properties of 1-motives.

Abstract

We describe algebraically defined cohomological and homological Albanese and Picard 1-motives (or mixed motives) of any algebraic variety in characteristic zero, generalizing the classical Albanese and Picard varieties. We compute Hodge, l-adic and De Rham realizations proving Deligne's conjecture for the concerned mixed Hodge structures. We investigate functoriality, universality, homotopical invariance and invariance under formation of projective bundles. We compare our cohomological and homological 1-motives for normal schemes. For proper schemes, we obtain an Abel-Jacobi map from the (Levine-Weibel) Chow group of zero cycles to our cohomological Albanese 1-motive which is the universal regular homomorphism to semi-abelian varieties. By using this universal property we get 'motivic' Gysin maps for projective local complete intersection morphisms. This paper is an extended version of our preliminary Comptes Rendus Note, Academie des Sciences, Paris, Vol. 326, 1998.