Abstract
We give an analytic description of the volume of a graded linear series, as the Monge–Ampère mass of a certain equilibrium metric associated to any smooth Hermitian metric on the line bundle. We also show the continuity of this equilibrium metric on some Zariski open subset, under a geometric assumption.
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Acknowledgments
The author would like to express his gratitude to his advisor Professor Shigeharu Takayama for his warm encouragements, suggestions and reading the drafts. The author is grateful to Atsushi Ito for several helpful comments on the algebraic reduction arguments in this paper. The author greatly indebted to the referee for his many kind advices on many mathematical or linguistic points of improvement. In particular he indicated to the author a rather straightforward proof of Theorem 3 after his reading the first version of this paper. In the first version the author used the uniform convergence in Proposition 21 to derive Theorem 3 but such a Bergman kernel estimate turned out to be unnecessary if one notes that \(\varphi _k\) is essentially non-decreasing. This research is supported by JSPS Research Fellowships for Young Scientists (22-6742).
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Hisamoto, T. On the volume of graded linear series and Monge–Ampère mass. Math. Z. 275, 233–243 (2013). https://doi.org/10.1007/s00209-012-1133-6
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DOI: https://doi.org/10.1007/s00209-012-1133-6