Nearly holomorphic functions on hermitian symmetric spaces
Nearly holomorphic functions on hermitian symmetric spaces
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厄米对称空间上的近全纯函数
DOI：
10.1007/bf01458058
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发表时间：
1987
影响因子：
1.4
通讯作者：
G. Shimura
中科院分区：
文献类型：
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作者：
G. Shimura
复Kfhler流形V上的近全纯函数的概念定义如下：取Ktihler形式t ′ f = i~？gq）上的一个函数，如果它是一个在r I中的多项式，则称它为近全纯函数. 3~ o/t~ z，具有全纯系数，其中{z I..... z，}是U上的一组复坐标函数。这类函数在我们以前的论文[16]中引入，目的是为了更清楚地理解某些自守定理，这些自守定理是非全纯的，但算术行为非常像全纯自守形式。这类级数自然地表现为Eisenstein级数，也表现为某些类似于它们的无穷级数。我们在[16]中的重点是这类级数几乎是全纯的，以及证明的方法。在本文中，我们更倾向于一般性，但在某些地方考虑了诸如Eisenstein级数的显式例子。虽然几乎全纯函数可以定义在任何复杂的Kfihler流形上，但埃尔米特对称空间似乎是它们最自然的栖息地。因此，在很大程度上，我们将我们目前的研究限制在这样的空间上的函数。
The notion of nearly holomorphic function on a complex Kfhler manifold V is defined as follows: Taking a Ktihler form t'/= i~? gq) on Vwith a real-valued function~ 0 in a coordinate neighborhood U, we call a function on Unearly holomorphic if it is a polynomial in r I..... 3~ o/t~ z, with holomorphic coefficients, where {z I..... z,} is a set of complex coordinate functions on U. This class of functions was introduced in our previous paper [16] for the purpose of gaining a clearer comprehension of certain automorphic lorms that are nonholomorphic but arithmetically behave very much like holomorphic automorphic forms. Such tbrms appear naturally as Eisenstein series and also as certain infinite series similar to them.Our emphasis in [16] was laid on thejact that such series are nearly holomorphic and also on the method of proof, in the present paper, we lean more towards generalities; yet explicit examples such as Eisenstein series are considered at some places. Though nearly holomorphic functions can be defined on any complex Kfihler manifold, hermitian symmetric spaces seem their most natural habitat. Therefore, lbr the most part, we restrict our present investigation to the functions on such spaces.