Abstract
Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \(\omega \) that has standard 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 Kähler metrics with singularities.
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Notes
Indeed one can show the error term is \(\varepsilon (k)\) since \(\omega \) has constant curvature, see for example [13].
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Acknowledgements
We would like to thank Professor Simon Donaldson for insightful discussions regarding quantization of Kähler metrics over long time, and we are grateful to Professors Xiuxiong Chen, Dror Varolin and Bin Xu for their interest in this result. This project started after the talk by the second author in the workshop “Quantum Geometry, Stochastic Geometry, Random Geometry, you name it” in the Simons Center in June 2015, and he thanks Steve Zelditch for the invitation. The first author would also like to thank Professor Xiuxiong Chen for the hospitality while his stay in USTC, and he is always grateful to Professor Bernard Shiffman for his continuous and unconditional support.
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Jingzhou Sun is partially supported by NNSF of China No. 11701353 and the STU Scientific Research Foundation for Talents No. 130/760181. Song Sun is partially supported by NSF Grant DMS-1405832 and Alfred P. Sloan fellowship.
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Sun, J., Sun, S. Projective Embedding of Log Riemann Surfaces and K-Stability. J Geom Anal 31, 5526–5554 (2021). https://doi.org/10.1007/s12220-020-00489-w
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DOI: https://doi.org/10.1007/s12220-020-00489-w