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Pull-back of currents by meromorphic maps. (Pull-back de courants par des applications méromorphes.) (English. French summary) Zbl 1335.32013

This paper establishes a new framework for the study of pull-backs of currents on compact Kähler manifolds by dominant meromorphic maps. The framework draws heavily from fairly disparate work by various authors (most substantially from work by T.-C. Dinh and N. Sibony [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 959–971 (2004; Zbl 1074.53058)]), and ultimately unifies and extends the pre-existing results.
Since the pre-existing results in this area of research are scattered widely, the introduction of the paper is quite useful in itself for a summary of the current state of the art as well as a precise explanation of the limitations of previous approaches. The main difficulty in pulling back a general \((p,p)\)-current \(T\) is the need to wedge a pull-back \(\pi^*T\) (where \(\pi\) is a morphism, so that \(\pi^*T\) is understood) with a current of integration. Previous approaches have typically computed this wedge product as a limit of wedge products of the current of integration with well-behaved currents approaching \(T\); the results then are limited to currents \(T\) for which convergent sequences of well-defined wedge products can be constructed. The new idea here is to define \(f^{\#}T\) indirectly as the unique current satisfying a suitable equality of integrals (which essentially amounts to an adjunction formula); this definition then depends on a different wedge product of currents, which can be defined in many more situations than achieved in previous work. Note that the new definition needs not succeed for all \((p,p)\)-currents; but it does have the advantage of making no a priori assumptions about \(T\).
A quite surprising result (Corollary 2) is that \(f^\#T\) can be non-positive for positive \(T\). This outcome can be viewed as the signature achievement of the new definition: previous methods could never exhibit this phenomenon for well-defined \(f^\#T\); but it now seems that the possibility of losing positivity is a natural consequence of defining pull-backs in optimal generality.
The proofs of the theorems showing that \(f^\#T\) has desirable properties and that \(f^\#T\) agrees with previously formulated notions require a strong technical understanding of DSH currents.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
32U40 Currents

Citations:

Zbl 1074.53058
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References:

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