On the Gauss-Manin connection of an isolated hypersurface singularity
On the Gauss-Manin connection of an isolated hypersurface singularity
复制标题
孤立超曲面奇点的高斯-马宁连接
DOI：
10.1007/bf01351450
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发表时间：
1978
影响因子：
1.4
通讯作者：
J. Scherk
中科院分区：
文献类型：
--
作者：
J. Scherk
假设f= f（x 0....，x1）是定义在一个开球中的解析函数，约为0，单位为IU”+ i。设f（0）= 0且0是唯一的临界点，Brieskorn研究了在B中奇点（f-1（0），0）的Gauss-Manin联络。他给了一个代数描述的连接，使他能够证明，本征值的monodromy的（f-1（0），0）是根的团结。本文对单值图的若当标准形中的块的大小作了一个猜想。通过计算实例，为这一猜想提供了一些证据。这是通过计算剩余的连接，事实上，这样做的方法是本文的主要结果。马尔格朗热的工作有关高斯-马宁连接渐近积分[M1]和伯恩斯坦多项式[M2]表明，能够计算的连接可能是非常有用的。在§ 2 Brieskorn的理论概述。§ 3讨论了单值定理，并提出了上述猜想。在§ 4中，给出了一个基，关于这个基，联络有一个简单的极点。§ 5描述了如何计算连接。在§ 6中，该方法对xs+ yS+ x2 y2进行。(This是第一个多项式被发现，其monodromy有无限阶，比照。[C].）我要感谢EV Brieskorn提供的许多建议和许多有益的讨论。
Suppose f= f (xo...., x,) is an analytic function defined in an open ball about 0 in IU"+ i. Assume that f (0)= 0 and that 0 is the only critical point off In B, Brieskorn studied the Gauss-Manin connection of the singularity (f-1 (0), 0). He gave an algebraic description of the connection which enabled him to prove that the eigenvalues of the monodromy of (f-1 (0), 0) are roots of unity. In this note we make a conjecture about the size of the blocks in the Jordan normal form of the monodromy. Some evidence for this conjecture has been gathered by calculating examples. This is done by computing a residue of the connection, and in fact a method for doing so is the principal result of this paper. Work by Malgrange relating the Gauss-Manin connection to asymptotic integrals [M1] and the Bernstein polynomial [M2] suggests that being able to calculate the connection may be quite useful.In § 2 Brieskorn's theory is sketched. § 3 discusses the monodromy theorem and makes the conjecture mentioned above. In § 4, a basis is given with respect to which the connection has a simple pole. § 5 describes how to calculate the connection. In § 6, this method is carried out for xs+ yS+ x2y2.(This was the first polynomial to be discovered, whose monodromy has infinite order, cf.[C].) I would like to thank EV Brieskorn for much advice and many helpful discussions.