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, Volume 152, Issue 1–2, pp 61–125 | Cite as

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

  • Luc IllusieEmail author


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.

Mathematics Subject Classification

Primary: 14F20 Secondary: 11T23 18F10 32S30 32S40 


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© © European Union 2016

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

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