Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part I
F. Reese Harvey, H. Blaine Lawson
备注：
Simplifications and revisions have been made in the proof of Theorem 6.1
发表信息：
in \The Many Facets of Geometry: a Tribute to Nigel Hitchin", J.-P. Bourguignon, O. Garcia-Prada, Simon Salamon, Eds., Oxford University Press, Oxford, England, 2008
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.
In Part I of this paper we introduced 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 of codimension p in P^n - M. It is shown here that a basic conjecture concerning the projective hull of real curves in P^2 implies the following result: M is the boundary of a positive holomorphic p-chain T of mass M(T) < K in P^n if and only if Link(M,Z) > -K p!deg(Z) for all algebraic subvarieties Z of codimension p in P^n - M. An analogous result is implied in any projective man…
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Wronskian curvature positivity is equivalent to the existence of a holomorphic projective connection
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
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Xiamen University Malaysia（厦门大学马来西亚分校）；Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；National Center for Theoretical Sciences（国家理论科学中心）；Department of Mathematics, National Taiwan University（台湾大学数学系）
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