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Published online by Cambridge University Press: 22 January 2016
A differential form co on a complete variety Un is said to be of the first kind if it is finite at every simple point of any variety which is birationally equivalent to U. Let k be a common field of definition for Uand ω, and let Pbe a generic point of Uover k. If ω is of the first kind, then ω(P) is of course a differential form of the first kind belonging to the extension k(P) of k.
1) If we omit the condition of perfectness this theorem does not hold in general.
2) If the problem of the reduction of the singularity over perfect field is solved affirmatively, this theorem is an immediate consequence of theorem 1 of S. Koizumi’s paper; On the differential forms of the first kind on algebraic varieties, Journal of the Mathematical Society of Japan, Vol. 2. However it would not be meaningless to give a simple direct proof.
3) See Kawahara, Y., On the differential forms on algebraic varieties, this journal, Vol. 4, Theorem 1Google Scholar.
4) This theorem has been proved also by S. Koizumi.
5) See A. Weil’s book, Foundations of Algebraic Geometry, Prop. 8 in Chapter IV.
6) See Prop. 4 of Koizumi’s paper loc. cit. 2).
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