Compact Rigid Analytic Spaces — With special regard to surfaces —
Compact Rigid Analytic Spaces — With special regard to surfaces —
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紧凑的刚性分析空间 - 特别考虑表面 -
DOI：
10.2969/aspm/01010765
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
K. Ueno
中科院分区：
文献类型：
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作者：
K. Ueno
刚性解析空间的概念是由Tate [T]引入的。从那时起，理论得到了相当大的发展，我们现在有了一个相当令人满意的一般理论（例如，适当的映射定理，GAGA，奇异性的解决方案）。因此，有多少已知的紧致复流形的结果与刚性解析空间的结果类似，这也许是值得一试的。下面我们将证明，紧致复流形上的许多重要结果在紧致光滑刚性解析空间范畴中都有对应的结果。例如，引入代数维数和科代拉维数的概念，给出代数约化和复正则映射的结构定理。利用这些一般结果，我们将发展与复解析曲面类似的紧致光滑刚性解析空间（刚性解析曲面）的双纯几何。刚性解析曲面自然地表现为曲面So在特征p> 0到特征零的形式提升的“一般”纤维S~。我们将证明S~和So的科代拉维数几乎在所有情况下都相等。(See定理5.11，如下。）我们猜想S~和So总是有相同的科代拉维数.如果我们知道代数维数为零的某些刚性解析曲面的结构，这将得到证明。目前，我们还没有关于这种表面的令人满意的理论。
The notion of rigid analytic spaces was introduced by Tate [T]. Since then the theory has been developed considerably and we now have a rather satisfactory general theory (for example, the proper mapping theorem, GAGA, resolution of singularities). Thus it may be worthwhile to see how many of the results known for compact complex manifolds have analogues for rigid analytic spaces. In the following we shall show that many important results on compact complex manifolds have counterparts in the category of compact smooth rigid analytic spaces. For example, the notions of algebraic dimension and Kodaira dimension will be introduced and the structure theorems of algebraic reductions and pluricanonical mappings will be shown. Using these general results, we shall develop the bimeromorphic geometry of compact smooth rigid analytic spaces of dimension two (rigid analytic surfaces) which is similar to that of the complex analytic surfaces. A rigid analytic surface appears naturally as the "generic" fibre S~ of a formal lifting of a surface So in characteristic p> 0 to characteristic zero. We shall show that the Kodaira dimensions of S~ and So are equal in almost all cases. (See Theorem 5.11, below.) We conjecture that S~ and So have always the same Kodaira dimension. This will be proved, if we know the structure of certain rigid analytic surfaces with algebraic dimension zero. At the moment, we have no satisfactory theory of such surfaces.