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Transactions of the American Mathematical Society

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Volumes of images of varieties in projective space and in Grassmannians


Author: H. Alexander
Journal: Trans. Amer. Math. Soc. 189 (1974), 237-249
MSC: Primary 32C30; Secondary 32E25
MathSciNet review: 0357850
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Abstract: If V is a complex analytic subvariety of pure dimension k in the unit ball in $ {{\mathbf{C}}^n}$ which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of $ (n - k)$-complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.


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  • [1] Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR 0054016
  • [2] H. Alexander, B. A. Taylor, and J. L. Ullman, Areas of projections of analytic sets, Invent. Math. 16 (1972), 335–341. MR 0302935
  • [3] S. S. Chern, Lectures on integral geometry, Notes by H. C. Hsiao, multilithed, 1965.
  • [4] G. de Rham, On currents in an analytic complex manifold, Seminars on Analytic Functions, vol. 1, Princeton, N.J., 1957, pp. 54-64.
  • [5] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
  • [6] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
  • [7] Fritz Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), no. 1, 1–88 (German). MR 1511365, 10.1007/BF01448415
  • [8] Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
  • [9] H. Kneser, Zur Theorie der gebrochenen Funktionen mehrer Veränderlichen, Jber. Deutsch. Math. Verein. 48 (1938), 1-28.
  • [10] Pierre Lelong, Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France 85 (1957), 239–262 (French). MR 0095967
  • [11] Toshio Nishino, Sur une propriété des familles de fonctions analytiques de deux variables complexes, J. Math. Kyoto Univ. 4 (1965), 255–282 (French). MR 0179384
  • [12] L. A. Santaló, Integral geometry in Hermitian spaces, Amer. J. Math. 74 (1952), 423–434. MR 0048062
  • [13] Wilhelm Stoll, About the convergence of a power series, Festschr. Gedächtnisfeier K. Weierstrasse, Westdeutscher Verlag, Cologne, 1966, pp. 523–529. MR 0197497
  • [14] Gabriel Stolzenberg, Volumes, limits, and extensions of analytic varieties, Lecture Notes in Mathematics, No. 19, Springer-Verlag, Berlin-New York, 1966. MR 0206337
  • [15] Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Tarrytown-on-Hudson, N. Y., 1971. MR 0423083

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0357850-9
Keywords: Complex analytic variety, volume of variety, Grassmann manifold, normal family, uniform algebra
Article copyright: © Copyright 1974 American Mathematical Society