University of Calgary

Seminar on nearby cycles - a PIMS Number Theory CRG Activity

Submitted by ccameron on Wed, 01/12/2011 - 3:34pm.
Jan 12 2011 - 3:05pm
Jan 12 2011 - 4:15pm
Speaker: 

Patrick Brosnan, Masoud Kamkar and guests via videoconference from UBC.

Location: 
UC BioSci 540B
http://www.math.ubc.ca/~masoud/nearby/

Our goal is to study the nearby cycles functor which is used to understand the cohomology of degenerating families of varieties. The nearby cycles functor has found many uses in algebraic geometry and its applications to representation theory and number theory, the most famous of which is Deligne's proof of Weil's Riemann hypothesis. However, the only comprehensive reference seems to be the original notes in SGA 7. Our aim is to go through these notes slowly and systematically to obtain a good grasp of the subject. Time permitting, we will also study some of the applications of this construction.

Brief mathematical introduction

Suppose f:XΔ is a morphism of complex-analytic spaces where Δ:={sC: |s|<1} denotes the unit disc in the complex plane. Let X0 denote the preimage of 0 in X and set X*:=X-X0. If the map f is suitably well-behaved, then the inclusion X0 X is a homotopy equivalence; thus, for any point ηΔ*:=Δ-{0}, there is a map Ψf:Xη X0 induced by the inclusion of Xη into X. Moreover, the diffeomorphism type of the generic fiber Xη does not depend on the choice of ηΔ*. With a little bit more thought, the nearby cycles map Ψf can even be defined in a canonical way without choosing a particular reference fiber Xη.

In many situations, Ψf can be interpreted geometrically as a map contracting certain vanishing cycles. In these cases, the Leray-Serre spectral sequence relates the cohomology of the generic fiber to that of the special fiber in a way that gives information about both.

The formalism of nearby cycles functors in SGA7 defines a suitable replacement for the map Ψf in the setting of étale cohomology. The first step is to replace the map Ψf, which is difficult to interpret algebraically, with the functor Rψf:Db (X*) Db (X0) between the bounded derived categories of sheaves on the two spaces.

The next step is to replace the disk Δ with the spectrum S=Spec R of a complete valuation ring R. In this setting, one lets s (resp. η) denote the closed (resp. generic) point in S. The nearby cycles functor Rψf:Db(Xη) Db (Xs) then sends constructible complexes on Xη to constructible complexes on Xs. This functor was first defined by the Grothendieck school in the early 70s in the framework of étale cohomology as the algebraic version of the analytic functor described above. Later, when perverse sheaves were invented, it became apparent that Rψf has the rather extraordinary property of sending perverse sheaves to perverse sheaves. Sometimes, for instance when f has finitely many critical points, this makes it possible to compute Rψf without having a detailed knowledge of f.

References

  • The main reference is SGA7, Expose XIII and XIV Pages 82 (=86) and 122 (=116), respectively.
  • Illusie's article on vanishing cycles.
  • Cataldo and Migliorini's article on the decomposition theorem has a nice section on nearby cycles.
  • Milnor's singular point on complex hypersurfaces.
  • Le Dung Trang: singularites isolees des hypersurfaces complexes.
  • For basics of l-adic cohomology, see Milne's notes.
  • History: Illusie's account of Grothendieck at IHES.

Schedule

  • January 12: Overview of the subject (Patrick Brosnan)
  • January 19: Galois theoretic description of sheaves on a variety - Section 1 of Ex. XIII (Masoud Kamkar)
  • February 26: Galois theoretic description of sheaves on a variety II - Section 1 of Ex. XIII (Masoud Kamkar)
  • February 2: Definition of nearby and vanishing cycles - Section 2 of Ex. XIII
  • February 9: Basic properties of nearby and vanishing cycles - Section 2 of Ex. XIII
  • February 16: Transcendental formalism - Section 1 of Ex. XIV
  • March 23: Comparison theorem - Section 2 of Ex. XIV
  • March 2: Isolated singularities - Section 3 of Ex. XIV
  • March 9: De Rham Cohomology - Section 4 of Ex. XIV
  • March 16:
  • March 23:
  • March 30: