Deligne pairings and the knudsen-mumford expansion

D.H. Phong Columbia University Julius Ross Columbia University Jacob Sturm Rutgers University

Differential Geometry mathscidoc:1609.10058

Journal of Differential Geometry, 78, (3), 475-496, 2008
Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(πLk) in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds for families of relative dimension one. As a corollary, we show that when X is smooth, the line bundle η associated to X → B, which was introduced in Phong-Sturm [12], coincides with the CM bundle defined by Paul-Tian [10, 11]. In a second and third corollaries, we establish asymptotics for the K-energy along Bergman rays generalizing the formulas obtained in [11].
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@inproceedings{d.h.2008deligne,
  title={DELIGNE PAIRINGS AND THE KNUDSEN-MUMFORD EXPANSION},
  author={D.H. Phong, Julius Ross, and Jacob Sturm},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909105605825044715},
  booktitle={Journal of Differential Geometry},
  volume={78},
  number={3},
  pages={475-496},
  year={2008},
}
D.H. Phong, Julius Ross, and Jacob Sturm. DELIGNE PAIRINGS AND THE KNUDSEN-MUMFORD EXPANSION. 2008. Vol. 78. In Journal of Differential Geometry. pp.475-496. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909105605825044715.
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