Luc Illusie - Around the Thom-Sebastiani theorem
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 03/10/2014

For germs of holomorphic functions $f : \mathbf{C}^{m+1} \to \mathbf{C}, g : \mathbf{C}^{n+1} \to \mathbf{C}$ 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 ⊕ 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), $x = (x_0,\cdots,x_m)$, $y = (y_0,\cdots,y_n)$. I will discuss algebraic variants and generalizations of this result over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. The main theorem is a Künneth formula for 𝑅𝛹 in the framework of Deligne's theory of nearby cycles over general bases.