Equidistribution results for singular metrics on line bundles
Equidistribution results for singular metrics on line bundles
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DOI：
10.24033/asens.2250
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发表时间：
2011-08
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通讯作者：
Dan Coman;G. Marinescu
中科院分区：
文献类型：
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作者：
Dan Coman;G. Marinescu
让L是一个全纯与正向弯曲奇异埃尔米特规线包在一个复杂的多方面的X可以定义自然的顺序Fubini-Study电流平方可积空间相关的全纯部分对张量的L .假设单数组中包含的指标是一个紧凑的分析X的子集,伯格曼内核函数的对数相关对张量的L(定义之外的奇异集)当p趋于无穷时，像o(p)一样增长，我们证明如下：1) k Fubini-Study电流弱收敛的力量在整个X k次方的曲率当前l . 2)共同的期望0平方可积的一个随机k-tuple全纯部分收敛弱的电流的k次方的曲率的l . k是这样奇异的余维数集的指标大于或等于k。弱渐近条件伯格曼内核函数在许多情况下，因为它是其渐近扩展的结果。我们也在一般情况下证明了它。然后我们证明了许多重要的几何情形（大线束上的奇异度量，zariski开集上的Kaehler-Einstein度量，算术商）适合我们的框架。
Let L be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents associated to the space of square integrable holomorphic sections of the p-th tensor powers of L. Assuming that the singular set of the metric is contained in a compact analytic subset of X and that the logarithm of the Bergman kernel function associated to the p-th tensor power of L (defined outside the singular set) grows like o(p) as p tends to infinity, we prove the following: 1) the k-th power of the Fubini-Study currents converge weakly on the whole X to the k-th power of the curvature current of L. 2) the expectations of the common zeros of a random k-tuple of square integrable holomorphic sections converge weakly in the sense of currents to to the k-th power of the curvature current of L. Here k is so that the codimension of the singular set of the metric is greater or equal as k. Our weak asymptotic condition on the Bergman kernel function is known to hold in many cases, as it is a consequence of its asymptotic expansion. We also prove it here in a quite general setting. We then show that many important geometric situations (singular metrics on big line bundles, Kaehler-Einstein metrics on Zariski-open sets, artihmetic quotients) fit into our framework.