Function theory on q-pseudoconvex domains

q-伪凸域上的函数论

• 批准号: 13640184
• 负责人: MATSUMOTO Kazuko
• 金额: $ 2.24万
• 依托单位: Osaka Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640184/
• 关键词: q-convex; levi form; plurisubharmonic; metric; several complex variables; 擬凸領域; Levi形式; 多変数関数論; 複素解析学; 多変数複素解析; q-convex; Levi平坦曲面

The results are the following:(1)The intersection of finite number of q-complete domains is cohomologically (q^^〜1)-complete if q does not devide n. This is best and shaper than that obtained by the approximation theorem of Diederich-Fornaess and the vanishing theorem of Andreotti-Grauert.(2)The complements of Levi flat real hypersurfaces in the complex tori of dimension 2 are Stein if they contain a local complex hypersurface which is not linear. This is the essential step to approach to the conjecture that there exists no Levi flat real hypersurfaces in complex tori except restricted types.(3)For a given complex submanifold M in C^n, we expressed the Hermitian matrix determined by the Levi form of the function -log d by using the defing function of M, where d denotes the distance to M. As its application, the degeneracy condition of the Levi form is equivalent to that of Gauss map.(4)For a given real submanifold M in C^2,we find a necessary and sufficient condition that the Levi form of distance to M degenerates. By applying this result, the Levi problem for complex tori can affimatively solved except the examples pointed out by Grauert.

结果表明：（1）有限个q-完备整环的交是上同调（q^^<$1）-完备的，如果q不整除n.这是最好的和形状比近似定理的Diederich-Fornaess和消失定理的Andreotti-Grauert。（2）Levi平坦真实的超曲面在2维复环面上的补曲面是Stein，如果它们包含一个非线性的局部复超曲面。这是证明复环面上除限制型外不存在Levi平坦真实的超曲面猜想的必要步骤。（3）对于C^n中的一个给定的复子流形M，我们用M的定义函数来表示由函数-log d的Levi形式所确定的Hermite矩阵，其中d表示到M的距离。作为其应用，Levi形式的退化条件与Gauss映射的退化条件等价。（4）对于C^2中给定的真实的子流形M，我们得到了到M的Levi形式距离退化的一个充要条件。除Grauert指出的例子外，应用这个结果，复环面上的Levi问题都能得到精确的解决。

项目成果

Kazuo Ishihara: "Iterative methods for eigenvalue probeems with nondefferentisble normalization condition of a general complex matrix"Computing Supplementum. 15. 105-118 (2001)

Kazuo Ishihara：“具有一般复数矩阵的不可微差归一化条件的特征值探针的迭代方法”计算补充。

Takashi Watanebe: "On the Bott suspension map for non-compact Lie groups"Journal of Mathematics of Kyoto University. 43. (2003)

Takashi Watanebe：“关于非紧李群的 Bott 悬浮图”京都大学数学学报。

Kouyemon Iriye: "The first derived functor of the inverse limit and localization"Journal of Pure and Applied Algebra. 173. 7-14 (2002)

Kouyemon Iriye：“第一个逆极限和局域化的导出函子”纯粹与应用代数杂志。

Kazuko Matsumoto: "Some geometric properties of Levi form of distance to real hypersurfaces in C^2"Japanese Journal of Mathematics. 30. (2004)

Kazuko Matsumoto：“C^2 中与真实超曲面距离的 Levi 形式的一些几何性质”日本数学杂志。

Kazuo Ishihara: "Descent iterations for improving approximate eigenpairs of polynomial eigenvalue problems with general complex matrices"Computing. 68. 239-254 (2002)

Kazuo Ishihara：“用于改进一般复矩阵多项式特征值问题的近似特征对的下降迭代”计算。