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

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 Y ⊂ X. For example, if E ≃ ℙⁿ⁻¹ is the exceptional divisor of a one-point blow-up X → X′, then T = [E] cannot be positively approximated: for every curve C ⊂ E, 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.