Star products of Green's currents and automorphic forms
Star products of Green's currents and automorphic forms
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格林电流和自同构形式的明星积
DOI：
10.1215/s0012-7094-01-10635-2
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发表时间：
2001
影响因子：
2.5
通讯作者：
J. Kramer
中科院分区：
文献类型：
--
作者：
J. Jorgenson;J. Kramer
在之前的工作中，作者计算了极化 n 维阿贝尔簇家族的埃尔米特线束的阿基米德高度。本文对阿贝尔纤维振动设置下获得的结果进行了详细分析，表明可以对证明进行修改，使其不再依赖于阿贝尔纤维振动的具体设置，从而扩展到相当一般的情况。具体来说，我们让 f : X → Y 是在某个基数上的任何平滑、射影、n 维复变簇族，并考虑 X 上配备平滑、埃尔米特度量的线丛。该数据与 Y 上的厄米线束 M 相关联，其特征在于第一陈类的条件。在温和的附加假设下，结果表明，对于 L 的一般选择部分，与这些部分相关的格林电流的 (n + 1) 倍星积的积分，沿着 f 的纤维积分，是 M 的全局部分的对数范数。此外，证明在某些一般设置中，M 的全局部分可以用原始部分的点评估来明确表示。当 M 是 2 度偏振的模空间时，这个一般结果的一个特别有趣的例子出现在偏振恩里克斯表面的设置中。在这种情况下，通过格林电流构造的全局截面等于 R. Borcherds 首次研究的 - 函数的幂。还提供了其他示例和问题。
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 : X → Y be any family of smooth, projective, ndimensional 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 -function first studied by R. Borcherds. Additional examples and problems are presented.
影响因子：
3.1
作者：
I. Dolgachev;I. Dolgachev
通讯作者：
I. Dolgachev;I. Dolgachev