Some Remarks on the Theory of Picard Varieties

Jun-ichi IGUSA

doi:10.2969/jmsj/00320345

Jun-ichi IGUSA

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1951 Volume 3 Issue 2 Pages 345-348

DOI https://doi.org/10.2969/jmsj/00320345

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Published: 1951 Received: October 30, 1951 Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Kyoto University

Right :
1) Kyôto University

Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Right : We shall use freely the results and terminology of Weil's book: Foundations of algebraic geometry, Am. Math. Soc. Colloq., Vol. 29 (1946).
1) See my paper, On the Picard varieties attached to algebraic varieties, Amer. J. of Math. Vol. 74 (1952). We cite this paper as (P).
2) Cf. (P) and also my paper, Algebraic correspondences between algebraic varieties, Jap. J. of Math., Vol. 3 (1951).
3) The V-divisor X(z) was defined in the following way. Let k be a field of definition of the continuous family X and let w be a generic point of A over k, then X(z) is one of the specializations of X(w)=p^γ_V[X·(w×V)] over the specialization w→z with reference to k.
4) Cf. W. L. Chow, On compact complex analytic varieties, Amer. J. of Math., Vol. 71 (1949), theorem 7.
5) This fact can be reduced easily to the case of curves. In this case the assertion is proved in Weil's book: Variétés Abéliennes et courbes algébriques, Act. Sc. et Ind. N° 1064 (1948), lemme 10.
6) Indeed this idea had been applied in my first (unpublished) proof of the birational invariance of the Picard variety.
7) I have borrowed this formulation (with a slight modification) from a letter of A. Weil to W. L. Chow in February 12, 1951. We note also that the functions ψ and ψ are essentially unique, which correspond to the “canonical function” (Cf. loc. cit. 5) in the case of curves.
8) An integral matrix M is called a complex multiplication from ω₁ to ω₂ if there exists a complex matrix μ such that μω₁=ω₂M.

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