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Speaker  Luc Illusie(Université Paris-Sud)

TitleNearby cycles over general bases and Thom-Sebastiani theorems

Time2016.1.8 14:00-16:00

2016.1.12 14:00-16:00

2016.1.15 14:00-16:00

2016.1.19 14:00-16:00

VenueN913

Abstract：

For germs of holomorphic functions f: Cm+1C, g: Cn+1C having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group Φm+n+1(fg) (and its monodromy) as a tensor product Φm(f)Φn(g), where (fg)(x,y)=f(x)+g(y), x=(x0,...,xm), y=(y0,...,yn). 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.

Plan：

1. Review of classical nearby and vanishing cycles

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

3. Ψ-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