A course on
Nearby cycles over general bases and Thom-Sebastiani theorems
by Prof. Luc Illusie

Date: Jan. 8 (Fri), Jan. 12 (Tue), Jan. 15 (Fri), Jan. 19 (Tue)
Time: 14:00—16:00
Place: South Building 913
Speaker: Prof.
Luc Illusie (Université Paris-Sud)

Title: Nearby cycles over general bases and Thom-Sebastiani theorems

Abstract: 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 $\Phi^{m+n+1}(f \oplus g)$ (and its monodromy) as a tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where $(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. 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 $R\Psi$ in the framework of Deligne's theory of nearby cycles over general bases.

Plan:

1. Review of classical nearby and vanishing cycles

2. Deligne's oriented products and nearby cycles over general bases

3. $\Psi$-goodness and Künneth theorems

4. Review of global and local additive convolution

5. Thom-Sebastiani type theorems

6. The tame case: monodromy and variation

7. Open questions

Reference: Around the Thom-Sebastiani theorem, my web page.