Boundaries of Positive Holomorphic Chains and the Relative Hodge Question

Details Boundaries of Positive Holomorphic Chains and the Relative Hodge Question Boundaries of Positive Holomorphic Chains and the Relative Hodge Question

2006-10-17

arxiv.org/abs/math/0610533v2

Mathematics

Complex Variables

New sections have been added on the relative Hodge question: Which relative homology classes in H_{2p}(X,M;Z) are represented

Scientific paper

We characterize the boundaries of positive holomorphic chains (with both compact and non-compact support) in an arbitrary complex manifold. We then consider a compact oriented real submanifold of dimension 2p-1 in a compact Kahler manifold X and address the question of which relative homology classes in H_{2p}(X,M;Z) are represented by positive holomorphic chains. Specifically, we define what it means for a class u in H_{2p}(X,M;Z) to be of type (p,p) and positive. It is then shown that u has these properties if and only if u = [T+S] where T is a positive holomorphic chain with dT = du and S is a positive (p,p)-current with dS = 0.

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