By Serge Lang (auth.)
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Extra info for Abelian Varieties
Putting u = 0, we obtain [II, § 1J 29 MAPS OF VARIETIES INTO ABELIAN VARIETIES d . x, which shows that the restriction of
H(P g ). It is symmetric in P v ... , P g , and the point v = H(Pv ... , P g ) is rational over k(u). Hence there exists a rational map (J. U = v. Taking into account Theorem 4, and recalling that I(P') = h(P') = 0, we see that (J. is a homomorphism. This proves our theorem. Let 1 : V ~ A be a rational map of a variety into an abelian variety. Then 1 induces a homomorphism of the group of cycles on V into A as follows. We denote by Zr(V) the group of cycles of dimension r on V. Let a = ~ ni(xi ) be an element of Zo(V)· We put I(a) = ~ ni(t(x i )).
We shall say that g* is the homomorphism induced by g. By an abuse of language, we shall sometimes say that A is an Albanese variety, and that I is a canonical map. THEOREM 11. Let V be a variety. Then there exists an Albanese variety (A, f) 01 V. The abelian variety A is uniquely determined up to a birational isomorphism, and I is determined up to a translation. Proof: The uniqueness of A and I up to a translation is an immediate consequence of the uniqueness of g* in (ii). To prove the existence, we note that the theorem is birational in V.