15 February 2001 Star products of Green's currents and automorphic forms
J. Jorgenson, J. Kramer
Duke Math. J. 106(3): 553-580 (15 February 2001). DOI: 10.1215/S0012-7094-01-10635-2


In previous work, the authors computed archimedian heights of hermitian line bundles on families of polarized, n-dimensional abelian varieties. In this paper, a detailed analysis of the results obtained in the setting of abelian fibrations is given, and it is shown that the proofs can be modified in such a way that they no longer depend on the specific setting of abelian fibrations and hence extend to a quite general situation. Specifically, we let f : XY be any family of smooth, projective, n-dimensional complex varieties over some base, and consider a line bundle on X equipped with a smooth, hermitian metric. To this data is associated a hermitian line bundle M on Y characterized by conditions on the first Chern class. Under mild additional hypotheses, it is shown that, for generically chosen sections of L, the integral of the (n+1)-fold star product of Green's currents associated to the sections, integrated along the fibers of f, is the log-norm of a global section of M. Furthermore, it is proven that in certain general settings the global section of M can be explicitly expressed in terms of point evaluations of the original sections. A particularly interesting example of this general result appears in the setting of polarized Enriques surfaces when M is a moduli space of degree-2 polarizations. In this setting, the global section constructed via Green's currents is equal to a power of the Φ-function first studied by R. Borcherds. Additional examples and problems are presented.


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J. Jorgenson. J. Kramer. "Star products of Green's currents and automorphic forms." Duke Math. J. 106 (3) 553 - 580, 15 February 2001. https://doi.org/10.1215/S0012-7094-01-10635-2


Published: 15 February 2001
First available in Project Euclid: 13 August 2004

zbMATH: 1045.11032
MathSciNet: MR1813236
Digital Object Identifier: 10.1215/S0012-7094-01-10635-2

Primary: 11F55
Secondary: 11F23 , 14G40 , 14J28 , 32N10

Rights: Copyright © 2001 Duke University Press


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Vol.106 • No. 3 • 15 February 2001
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