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The Many Facets of GeometryA Tribute to Nigel Hitchin$
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Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon

Print publication date: 2010

Print ISBN-13: 9780199534920

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199534920.001.0001

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Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part I

Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part I

Chapter:
(p.261) XIII Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part I
Source:
The Many Facets of Geometry
Author(s):

F. Reese Harvey

H. Blaine Jr. Lawson

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199534920.003.0013

In 2000, H. Alexander and J. Wermer published the following result: Let Γ be a compact-oriented smooth submanifold of dimension 2p - 1 in ℂn. Then Γ bounds a positive holomorphic p-chain in ℂn if and only if the linking number Link(Γ, Z) ≥ 0 for all canonically oriented algebraic subvarieties Z of codimension p in ℂn - Γ. This chapter formulates and proves an analogue of the Alexander–Wermer theorem for oriented (not necessarily connected) curves in a projective manifold.

Keywords:   Alexander–Wermer theorem, oriented curves, projective manifold, H. Alexander and J. Wermer

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