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

Details 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

2005-12-15

arxiv.org/abs/math/0512379v4

Mathematics

Complex Variables

Simplifications and revisions have been made in the proof of Theorem 6.1

Scientific paper

We introduce the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z in P^n - M of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it generalizes directly from P^n to any projective manifold. Part 1 of this paper establishes the following result for the case p=1. Let M be an oriented, stable, real analytic curve in P^n. Then M is the boundary of a positive holomorphic 1-chain T with Mass(T) < K in P^n if and only if Link(M,Z) > -K deg(Z) for all algebraic hypersurfaces Z in P^n - M. An analogous theorem is implied in any projective manifold. Part 2 of this paper studies similar results for p>1.

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