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Regularization of closed positive currents of type (1,1) by the flow of a Chern connection

  • Jean-Pierre Demailly
Chapter
Part of the Aspects of Mathematics book series (ASMA, volume E 26)

Abstract

Let X be a compact n-dimensional complex manifold and let T be a closed positive current of bidegree (1, 1) on X. In general, T cannot be approximated by closed positive currents of class C : a necessary condition for this is that the cohomology class {T} is numerically effective in the sense that ∫ Y {T} P ≥ 0 for every p-dimensional subvariety YX. For example, if E ≃ ℙ n−1 is the exceptional divisor of a one-point blow-up XX′, then T = [E] cannot be positively approximated: for every curve CE, we have ∫ C {E} = ∫ C c 1(O(−1)) < 0. However, we will see that it is always possible to approximate a closed positive current T of type (1, 1) by closed real currents admitting a small negative part, and that this negative part can be estimated in terms of the Lelong numbers of T and the geometry of X.

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© Springer Fachmedien Wiesbaden 1994

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  • Jean-Pierre Demailly

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