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RC-positivity and scalar-flat metrics on ruled surfaces

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Abstract

Let X be a ruled surface over a curve of genus g. We prove that X has a scalar-flat Hermitian metric if and only if \(g\ge 2\) and \(m(X)>2-2g\) where m(X) is an intrinsic number depending on the complex structure of X. As an application, we construct explicit examples of compact Kähler manifolds which have scalar-flat Hermitian metrics, but can not support scalar-flat Kähler metrics. We also classify compact complex surfaces with scalar-flat Hermitian metrics.

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Acknowledgements

The first author would like to thank his advisor Professor Jian Zhou for his guidance. The second author is very grateful to Professor K.-F. Liu and Professor S.-T. Yau for their support, encouragement and stimulating discussions over years. We would also like to thank Professors A. Futaki, S. Sun, G. Szekelyhidi, V. Tosatti, J. Xiao, W.-P. Zhang and X.-Y. Zhou for some helpful discussions

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Authors and Affiliations

  1. Shing-Tung Yau Center of Southeast University, Nanjing, 211189, China

    Jun Wang

  2. Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Xiaokui Yang

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  1. Jun Wang
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Correspondence to Xiaokui Yang.

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This work was partially supported by China’s Recruitment Program of Global Experts, NSFC12141101 and NSFC12171262.

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Wang, J., Yang, X. RC-positivity and scalar-flat metrics on ruled surfaces. Math. Z. 301, 917–934 (2022). https://doi.org/10.1007/s00209-021-02934-0

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