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Regularization of currents and entropy. (English) Zbl 1074.53058

Authors’ summary: Let \(T\) be a positive closed \((p,p)\)-current on a compact Kähler manifold \(X\). We prove the existence of smooth positive closed \((p,p)\)-forms \(T_ n^ +\) and \(T_ n^ -\) such that \(T_ n^ + - T_ n^ -\to T\) weakly. Moreover, \(\| T_ n^ \pm\| \leq c_ X\| T\| \) where \(c_ X >0\) is a constant independent of \(T\). We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed \((1,1)\)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate for the topological entropy of meromorphic maps on compact Kähler manifolds.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32U40 Currents
32C30 Integration on analytic sets and spaces, currents
37B40 Topological entropy
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H04 Meromorphic mappings in several complex variables
32Q15 Kähler manifolds
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