Generalized albanese and its dual

Abstract

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic 0.We consider categories of rational maps from $X$ to commutative algebraic groups, and ask for objects satisfying the universal mapping property.A necessary and sufficient condition for the existence of such universal objects is given, as well as their explicit construction, using duality theory of generalized 1-motives. An important application is the Albanese of a singular projective variety, which was constructed by Esnault, Srinivas and Viehweg as a universal regular quotient of a relative Chow group of 0-cycles of degree 0 modulo rational equivalence.We obtain functorial descriptions of the universal regular quotient and its dual 1-motive.