Factor sets and differentials on abelian varieties

Abstract

The main results of this paper are stated in 2.8 and 3.7; the notations and language are those used by the author in his previous work; numbers in brackets refer to the bibliography at the end of the paper. Although a few results of ??1 and 2 are valid for any characteristic, this paper intends to treat only the case of characteristic zero. The case of positive characteristic is treated in a forthcoming paper, since it needs an analysis of the derivations of higher order, and is connected to the rather surprising fact that an abelian variety over a field of positive characteristic may very well possess exact differentials of the first kind. ??1 and 2 contain the proofs of those properties of the differentials of the second kind which are needed in ?3; these properties are familiar in classical algebraic geometry, but their algebraic proofs are new. While our main interest, in ??1 and 2, rests with the differentials of the second kind, certain properties of the differentials of the first kind are also found, usually as special cases; the algebraic proofs of most of these properties are not new, and can be found also in [S. Koizumi, On the differentialforms of the first kind on algebraic varieties, J. Math. Soc. Japan vol. 1 (1949) p. 273] and [S. Nakano, On invariant differential forms on group varieties, ibid. vol. 2 (1951) p. 216]. 1. The differentials of the first and second kind. Let V be an irreducible variety over the field k; let W be an irreducible subvariety of V, and set Q=Q(W/V), $1= $3(W/V); let D be a derivation on V (see ?5 of [4]). We shall say that D is regular at W if Dx E Q when x E Q, and Dx (=-3 when x (= . If, in addition, DxEG when xCQ, we shall say that D has a zero at W. Let '7r be the homomorphic mapping of Q into k(W) whose kernel is 3; if D is regular at W, for xGQ we see that '7r(Dx) depends only on '7rx, and from this follows the existence of a derivation 7rD on W such that (7rD) ('7rx) = '7r(Dx) for x E Q; rD is called the derivation induced by D on W; clearly, 7rD=0 if and only if D has a zero at W.