学术报告：Nearby cycles over general bases and Thom-Sebastiani theorems

中科院数学与系统科学研究院

数学研究所

中科院晨兴数学中心

学术报告会

报告人：  Luc Illusie(Université Paris-Sud)

题  目：Nearby cycles over general bases and Thom-Sebastiani theorems

时  间：2016.1.8（星期五）， 14:00-16:00

2016.1.12（星期二）， 14:00-16:00

2016.1.15（星期五）， 14:00-16:00

2016.1.19（星期二）， 14:00-16:00

地  点：数学院南楼N913室

摘要：

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

计划：

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

参考文献：

Luc Illusie, Around the Thom-Sebastiani theorem, preprint.

Lectures co-sponsored by YMSC, Tsinghua University.