Abstract
Let L be a negative line bundle on a compact complex manifold Y, we define a Schwarz-type symmetrization on the total space X of L. We prove that this symmetrization does not increase the Monge–Ampère energy for fibrewise \(S^{1}\)-invariant plurisubharmonic functions defined on the ”unit ball” in X. As a result, we generalize the sharp Moser–Trudinger inequality to this ”unit ball”.
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Acknowledgements
This note was finished when the author was visiting Chalmers. He would like to thank Professor Bo Berndtsson, who encouraged him to consider this problem and provided many useful suggestions. Thanks also go to his supervisor Professor Jixiang Fu, who provided support for his visit. Finally, he is grateful for the referee for a detailed reading and many suggestions for improvement, especially for pointing out one mistake in the proof of Theorem 1.1 in the original manuscript.
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Wu, J. Symmetrization of plurisubharmonic functions and Fano manifolds. Math. Z. 295, 1553–1568 (2020). https://doi.org/10.1007/s00209-019-02393-8
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DOI: https://doi.org/10.1007/s00209-019-02393-8