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A geometric approach to problems in birational geometry

Chen-Yu Chi Shing-Tung Yau

Algebraic Geometry mathscidoc:1912.43593

Proceedings of the National Academy of Sciences, 105, (48), 18696-18701, 2008.12
A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: Given two mildly singular projective varieties with some of the first variety9s pseudonormed spaces being isometric to the corresponding ones of the second variety9s, can one construct a birational map between them that induces these isometries? In this work, a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.
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@inproceedings{chen-yu2008a,
  title={A geometric approach to problems in birational geometry},
  author={Chen-Yu Chi, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204314947115157},
  booktitle={Proceedings of the National Academy of Sciences},
  volume={105},
  number={48},
  pages={18696-18701},
  year={2008},
}
Chen-Yu Chi, and Shing-Tung Yau. A geometric approach to problems in birational geometry. 2008. Vol. 105. In Proceedings of the National Academy of Sciences. pp.18696-18701. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204314947115157.
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