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A cut-off theorem for plurisubharmonic currents. (English) Zbl 0808.32010
The paper continues the study of plurisubharmonic (psh) currents begun by L. Alessandrini and the author in Forum Math. 5, No. 6, 577-602 (1993; Zbl 0784.32014). A current \(T\) on a domain \(\Omega \subset \mathbb{C}^ N\) is said to be psh if \(i \partial \overline \partial T \geq 0\). It was shown by Sibony [N. Sibony, Duke Math. J. 52, 157-197 (1985; Zbl 0578.32023)] that for any positive psh current \(T\) of bidimension \((p,p)\) such that \(dT\) has measure coefficients and for every analytic subset \(Y\) of \(\Omega\) of pure dimension \(p\), there exists a weakly psh function \(f\) on \(Y\) such that \(\chi_ YT = f[Y]\) (where \(\chi_ Y\) denotes the characteristic function of \(Y)\). The assumption about \(dT\) is equivalent to require that \(T\) is normal and seems to be restrictive. The main result of the paper under review is avoiding these restrictions in Sibony’s theorem. To do this the author introduces the notions of \(\mathbb{C}\)-flat and \(\mathbb{C}\)-normal currents: a current \(T\) is said to be \(\mathbb{C}\)-flat if it can be expressed as \(F + \partial G + \overline \partial H\) where \(F,G\) and \(H\) have locally summable coefficients, and it is said to be \(\mathbb{C}\)-normal if both \(T\) and \(\partial \overline \partial T\) have measure coefficients. The theory of such currents is developed analogous to the classical one (of normal and flat currents).

MSC:
32C30 Integration on analytic sets and spaces, currents
32U05 Plurisubharmonic functions and generalizations
58A25 Currents in global analysis
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