Around the Thom–Sebastiani theorem, with an appendix by Weizhe Zheng

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• 作者：Luc Illusie
• 关键词：Mathematics Subject ClassificationPrimary ; 14F20 ; Secondary ; 11T23 ; 18F10 ; 32S30 ; 32S40
• 刊名：manuscripta mathematica
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：152
• 期：1-2
• 页码：61-125
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1785
• 卷排序：152

文摘

For germs of holomorphic functions \(f: (\mathbf {C}^{m+1},0) \rightarrow (\mathbf {C},0)\), \(g: (\mathbf {C}^{n+1},0) \rightarrow (\mathbf {C},0)\) having an isolated critical point at 0 with value 0, the classical Thom–Sebastiani theorem describes the vanishing cycles group \(\Phi ^{m+n+1}(f \oplus g)\) (and its monodromy) as a tensor product \(\Phi ^m(f) \otimes \Phi ^n(g)\), where \((f \oplus g)(x,y) = f(x) + g(y), x = (x_0,{\ldots },x_m), y = (y_0,{\ldots },y_n)\). We prove algebraic variants and generalizations of this result in étale cohomology over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. They generalize Fu (Math Res Lett 21:101–119, 2014). The main ingredient is a Künneth formula for \(R\Psi \) in the framework of Deligne’s theory of nearby cycles over general bases. In the last section, we study the tame case, and the relations between tensor and convolution products, in both global and local situations.