Holomorphic curves in surfaces of general type

SS Lu Shing-Tung Yau

Complex Variables and Complex Analysis mathscidoc:1912.43535

Proceedings of the National Academy of Sciences, 87, (1), 80-82, 1990.1
This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy c(2)1>2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a finite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q - 1 irrespective of the map.
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@inproceedings{ss1990holomorphic,
  title={Holomorphic curves in surfaces of general type},
  author={SS Lu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203903907241099},
  booktitle={Proceedings of the National Academy of Sciences},
  volume={87},
  number={1},
  pages={80-82},
  year={1990},
}
SS Lu, and Shing-Tung Yau. Holomorphic curves in surfaces of general type. 1990. Vol. 87. In Proceedings of the National Academy of Sciences. pp.80-82. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203903907241099.
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