The universal regular quotient of the Chow group of points on projective varieties

Abstract

Let X be a projective variety of dimension n de®ned over an algebraically closed ®eld k. For X irreducible and non-singular, Matsusaka [Ma] constructed an abelian variety Alb X † and a morphism a : X ! Alb X † (called the Albanese variety and mapping respectively), depending on the choice of a base-point on X , which is universal among the morphisms to abelian varieties (see Lang [La], Serre [Se] for other constructions). Over the ®eld of complex numbers the existence of Alb X † and a was known before, and has a purely Hodge-theoretic description (see Igusa [I] for the Hodge theoretic construction). Incidentally, the terminology ``Albanese variety'' was introduced by A. Weil, for reasons explained in his commentary on the article [1950a] of Volume I of his collected works (see [W]), one of which is that the paper [Alb] of Albanese de®nes it (for a surface) as a quotient of the group of 0-cycles of degree 0 modulo an equivalence relation. Let CHn X †deg 0 denote the Chow group of 0-cycles of degree 0 on X modulo rational equivalence. When X is irreducible and non-singular, a remarkable feature of the Albanese morphism a is that it factors through a regular homomorphism u : CHn X †deg 0 ! Alb X †, that is a homomorphism, which when composed with the cycle map c : X ! CHn X †deg 0, gives an algebraic morphism. This follows immediately from the fact that an abelian variety does not contain any rational curve. Thus one can reformulate Matsusaka's theorem as the Invent. math. 135, 595±664 (1999) DOI 10.1007/s002229900890