Intrinsic capacities on compact Kähler manifolds. (English) Zbl 1087.32020

The authors develop a global pluripotential theory in the context of compact Kähler manifolds. From the maximum principle it follows that there are no plurisubharmonic (psh) functions on compact complex manifolds. If \(\omega\) is a real closed smooth form of bidegree \((1,1)\) on the complex manifold \(X\), one can obtain positive closed currents \(\omega^\prime\) of bidegree \((1,1)\) which are cohomologous to \(\omega\). If \(X\) is Kähler we can write \(\omega^\prime =\omega +dd^c\varphi\), where \(\varphi\) is a \(\omega\)-plurisubharmonic (\(\omega\) -psh)function (or a quasiplurisubharmonic (qpsh) function). The authors define \(\omega\)-psh functions and gather useful facts about them, define the complex Monge-Ampère operator, establish Chern-Levine-Nirenberg inequalities, and study the Monge-Ampère capacity. Next they study the Alexander capacity and show that locally pluripolar sets can be defined by \(\omega\) psh functions when \(\omega\) is Kähler.


32U15 General pluripotential theory
32U05 Plurisubharmonic functions and generalizations
32Q15 Kähler manifolds
31C10 Pluriharmonic and plurisubharmonic functions
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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