The weighted Monge-Ampère energy of quasiplurisubharmonic functions. (English) Zbl 1143.32022

Guedj, Vincent; Zeriahi, Ahmed

J. Funct. Anal. 250, No. 2, 442-482 (2007).

Over the last few decades, complex Monge-Ampére equations have played a crucial role in Kähler geometry and complex dynamics. While, in the smooth case, the existence of a solution was successfully established by S. T. Yau [Comm. Pure Appl. Math. 31, 339-411 (1978; Zbl 0369.53059)], it is a very useful problem to study the behaviour of such equation in the singular case.
In the paper under review, the authors establish very interesting results in this direction. More specifically, let \(X\) be a compact Kähler manifold of dimension \(n\), with Kähler form \(\omega\). A function \(\phi\) is called \(\omega\)-plurisubharmonic if the current \(\omega_\phi=\omega+d d^c\phi\) is positive. Given a Radon measure \(\mu\) on \(X\), \(\phi\) is a solution of the complex Monge-Ampére equation if \(\omega_\phi^n=\mu\). The authors study a new class \(\mathcal E(X,\omega)\) of \(\omega\)-plurisubharmonic functions for which \(\omega_\phi^n\) is well defined. In particular, this is the largest class on which the comparison principle is valid. If \(X\) is a compact Riemann surface, then the set \(\mathcal E(X,\omega)\) corresponds to the set of \(\omega\)-subharmonic functions whose Laplacian does not charge polar sets. In general, it is shown that the Monge-Ampére equation \(\omega_\phi^n=\mu\) admits a solution \(\phi\in \mathcal E(X,\omega)\) if and only if \(\mu\) does not charge pluripolar sets.
The authors also obtain some uniqueness properties, generalizing Calabi’s result to the singular case, and derive some applications to complex dynamics and to the existence of singular Kähler-Einstein metrics.

Reviewer: Paolo Cascini (Santa Barbara)

MSC:

32W20

Complex Monge-Ampère operators

Keywords:

Complex Monge-Ampère operator; Kähler manifold; \(\omega\)-plurisubharmonic functions; weighted energy

Citations:

Zbl 0369.53059

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