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Curvature of the base manifold of a Monge–Ampère fibration and its existence

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Abstract

In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge–Ampère equation, which is called a Monge–Ampère fibration. There exist two canonical types of generalized Weil–Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil–Petersson metric, we obtain an explicit curvature formula and prove that the holomorphic bisectional curvature is non-positive, the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature are all bounded from above by a negative constant. For a holomorphic vector bundle over a compact Kähler manifold, we prove that it admits a projectively flat Hermitian structure if and only if the associated projective bundle fibration is a Monge–Ampère fibration. In general, we can prove that a relative Kähler fibration is Monge–Ampère if and only if an associated infinite rank Higgs bundle is Higgs-flat. We also discuss some typical examples of Monge–Ampère fibrations.

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Notes

  1. The name of the Monge–Ampère fibration was firstly given by Professor Bo Berndtsson.

  2. The negativity results of curvature are also obtained by Professor Bo Berndtsson independently using a different method based on the holomorphic motion structure of the fibration (see [7]).

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Acknowledgements

We would like to thank Bo Berndtsson and Ya Deng for several useful discussions about the topics of this paper. We also would like to thank the anonymous reviewers for their comments that helped improve the paper.

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  1. Mathematical Science Research Center, Chongqing University of Technology, Chongqing, 400054, China

    Xueyuan Wan

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Xu Wang

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  1. Xueyuan Wan
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Xueyuan Wan is partially supported by the National Natural Science Foundation of China (Grant No. 12101093) and Scientific Research Foundation of the Chongqing University of Technology.

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Wan, X., Wang, X. Curvature of the base manifold of a Monge–Ampère fibration and its existence. Math. Ann. 387, 353–387 (2023). https://doi.org/10.1007/s00208-022-02475-9

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