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Projective embedding of log Riemann surfaces and K-stability

Jingzhou Sun SONG SUN

Differential Geometry mathscidoc:1912.43459

arXiv preprint arXiv:1605.01089, 2016.5
Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \omega with cusp singularities along a divisor D, we show the L^ 2 projective embedding of (X, D) defined by L^ k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Khler metrics with singularities.
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@inproceedings{jingzhou2016projective,
  title={Projective embedding of log Riemann surfaces and K-stability},
  author={Jingzhou Sun, and SONG SUN},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114551544737019},
  booktitle={arXiv preprint arXiv:1605.01089},
  year={2016},
}
Jingzhou Sun, and SONG SUN. Projective embedding of log Riemann surfaces and K-stability. 2016. In arXiv preprint arXiv:1605.01089. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114551544737019.
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