Quasi-Albanese Variety

The quasi-Albanese variety of a connected smooth quasi-projective variety with a base point is the semiabelian variety receiving its universal based algebraic morphism . Every based algebraic map from to a semiabelian variety factors uniquely through it by a group homomorphism. Universality concerns algebraic morphisms, rather than arbitrary holomorphic maps. For a smooth projective variety it reduces to the Albanese variety. For based at 1, the map to is the identity.