Regularization of closed positive currents and intersection theory. (English) Zbl 0777.32016

Demailly, Jean-Pierre

J. Algebr. Geom. 1, No. 3, 361-409 (1992).

Let \(X\) be a compact complex manifold and let \(T=i\partial\overline\partial\psi\) be a closed positive current of bidegree (1,1) on \(X\). Under some hypothesis on a lower bound for the Chern curvature of the tangent bundle \(TX\), the current \(T\) is proved to be the weak limit of closed currents \(T_ k={i\over\pi}\partial\overline\partial\psi_ k\) with controlled negative parts; the functions \(\psi_ k\) decrease to \(\psi\) as \(k\to\infty\) and can be chosen smooth on \(X\). However the presence of positive Lelong numbers of \(T\) results in some loss of positivity of \(T_ k\).
This regularization is applied to relations between effective and numerically effective divisors, and to some problems of intersection theory.

Reviewer: A.Yu.Rashkovsky (Khar’kov)

MSC:

32J25	Transcendental methods of algebraic geometry (complex-analytic aspects)
32C30	Integration on analytic sets and spaces, currents
32S60	Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)

Keywords:

Lelong number; closed positive current; intersection theory