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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0357850-9
PII: S 0002-9947(1974)0357850-9
Keywords: Complex analytic variety, volume of variety, Grassmann manifold, normal family, uniform algebra
Article copyright: © Copyright 1974 American Mathematical Society