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Complex Monge–Ampère equations on quasi-projective varieties

Eleonora Di Nezza and Chinh H. Lu

Abstract

We introduce generalized Monge–Ampère capacities and use these to study complex Monge–Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor. Our results still hold if the divisor is replaced by any closed subset.

Funding statement: The authors are partially supported by the French ANR project MACK. The second-named author is supported by the European Research Councils.

Acknowledgements

It is our pleasure to thank our advisors Vincent Guedj and Ahmed Zeriahi for providing constant help, many suggestions and encouragements. We also thank Robert Berman and Bo Berndtsson for very useful comments. We are indebted to Sébastien Boucksom and Henri Guenancia for a very careful reading of a preliminary version of this paper, for their suggestions which improve the presentation of the paper. We would like to thank the referee for many helpful comments.

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Received: 2014-2-2
Revised: 2014-7-9
Published Online: 2014-9-26
Published in Print: 2017-6-1

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