Indices of 1-forms on an isolated complete intersection singularity
Indices of 1-forms on an isolated complete intersection singularity
复制标题
孤立完全交集奇点上 1-形式的指数
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
S. Gusein
中科院分区：
文献类型：
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作者：
W. Ebeling;S. Gusein
经典的Eisenbud-Levine-Khimshashvili公式在R^n$上的解析向量场的奇点指数上有一些推广。我们提供了一种替代方法的基础上的研究指标的1-形式，而不是向量场。当簇是一个真实的孤立的完全交奇点时，我们定义了它的（真实的）1-形式的指标，在复情形下，我们定义了复奇点的全纯1-形式的指标，并将其表示为某个代数的维数。在真实的设置中，对于一个N $V=f^{-1}（0）$，$f：（C^n，0） o（C^k，0）$，$f$是真实的，我们定义一个由映射$f$的像$（C^k，0）$的点$f $参数化的二次型的复解析族，对于真实的$f $，这些点成为真实的，在这种情况下，它们的签名从“真实的”索引推迟$x（V_∞）-1 $，其中$chi（V_∞）$是对应的平滑$V_∞ =f^{-1}（∞）帽B_delta$的欧拉特征线。
There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an icis $V=f^{-1}(0)$, $f:(C^n, 0) o (C^k, 0)$, $f$ is real, we define a complex analytic family of quadratic forms parameterized by the points $epsilon$ of the image $(C^k, 0)$ of the map $f$, which become real for real $epsilon$ and in this case their signatures defer from the "real" index by $chi(V_epsilon)-1$, where $chi(V_epsilon)$ is the Euler characteristic of the corresponding smoothing $V_epsilon=f^{-1}(epsilon)cap B_delta$ of the icis $V$.