Oka’s inequality for currents and applications. (English) Zbl 0832.32010

A version of Oka’s inequality for currents of the form \(uT\) is given, where \(T\) is a positive closed current of bidimension \((p,p)\) and \(u\) is a negative plurisubharmonic function. It follows that the set where \(uT\) has a locally bounded mass is \(p\)-pseudoconvex (using a convention where \((n - 1)\)-pseudoconvexity is the usual pseudoconvexity).
When \(uT\) has a locally bounded mass a convergence result is proved for the operator \[ (u_1 \ldots u_p) \to dd^cu_1 \wedge \cdots \wedge dd^c u_p \wedge T. \] This leads to a Bezout type theorem in \(\mathbb{P}^k\) with an application to holomorphic dynamics in \(\mathbb{P}^k\).


32C30 Integration on analytic sets and spaces, currents
32F10 \(q\)-convexity, \(q\)-concavity
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F99 Dynamical systems over complex numbers
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