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Published online by Cambridge University Press: 22 January 2016
In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).
1) We refer this book by F in this note.
2) Koizumi, S., On the differential forms of the first kind on algebraic varieties. I. Journal of the Mathematical Society of Japan, vol. 1 (1949). II. vol. 2 (1951)Google Scholar.
3) See Zariski, O., The reduction of the singularities of an algebraic surface. Annals of Math. vol. 40 (1939)CrossRefGoogle Scholar.
4) Loc. cit. 2).
5) Loc. cit. 3).
6) Zariski, O., Reduction of singularities of algebraic three-dimensional varieties, Annals of Math. vol. 45 (1944)CrossRefGoogle Scholar.
7) Loc. cit 2) S. Koizumi I. Prop. 6.
8) Even if ω is of the first kind, this is not always true.
9) Schmidt, F. K., Zur arithmetischen Theorie der algebraischen Funktionen II, § 5. Math. Zeitschrift, Bd. 35 (1939)Google Scholar.
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