Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds

We introduce the notion of the projective linking number LinkP(Γ, Z) of a compact oriented real submanifold Γ of dimension 2p − 1 in complex projective n-space P with an algebraic subvariety Z ⊂ P − Γ of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it is generalized to any projective manifold X . It is shown that a basic conjecture concerning the projective hull of a real analytic curve in P 2 implies the following result: Γ is the boundary of a positive holomorphic p-chain of mass ≤ Λ in P if and only if the L̃inkP(Γ, Z) ≥ −Λ for all algebraic subvarieties Z ⊂ P −Γ. An analogous result is implied in any projective manifold X .