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Pull-back of currents by holomorphic maps. (English) Zbl 1128.32020
Motivated by potential applications in complex dynamics the authors seek for the right definition of the pull-back of a positive current via a holomorphic map under possibly weak assumptions on the map. They prove that if \(F:X\to X'\) is holomorphic between two complex manifolds with fibres which are either empty or analytic of dimension \(\dim X - \dim X'\) then the pull-back \(f^{\ast }\) can be continuously extended from smooth \((p,p) \) forms to positive (resp., \(dd^c \) - closed) currents so that the positivity (resp., closedness) are preserved. If \(T\) puts no mass on a Borel set then \(f^{\ast }T\) has no mass on the preimage of the set. If, moreover \(X, X'\) are Kähler and compact then \(f^{\ast }\) preserves also cohomology classes: \(f^{\ast }\{ T\} = \{ f^{\ast }T\}\). This result is then extended to more general currents, the class denoted by DSH, which can be represented as a difference of two negative \((p,p)\) currents and their \(dd^c \) differential is a difference of two positive closed currents.
One can also consider more general mappings meromorphic ones or so called meromorphic transforms introduced by the same authors [T.-C. Dinh and N. Sibony, Comment. Math. Helv. 81, 221–258 (2006; Zbl 1094.32005)].
For the proofs the slicing of currents method is used and a generalized version of Chern-Levine-Nirenberg inequalities which works for currents which are not closed. The latter is also applied to prove Skoda-type extension theorem for such currents.

MSC:
32U40 Currents
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32H04 Meromorphic mappings in several complex variables
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