Seminar on nearby cycles - a PIMS Number Theory CRG Activity

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:X→Δ is a morphism of complex-analytic spaces where Δ:={s∈C: |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 η∈Δ^*:=Δ-{0}, there is a map Ψ_f:X_η→ X₀ induced by the inclusion of X_η into X. Moreover, the diffeomorphism type of the generic fiber X_η does not depend on the choice of η∈Δ^*. With a little bit more thought, the nearby cycles map Ψ_f can even be defined in a canonical way without choosing a particular reference fiber X_η.

In many situations, Ψ_f 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 Ψ_f in the setting of étale cohomology. The first step is to replace the map Ψ_f, which is difficult to interpret algebraically, with the functor Rψ_f:D^b (X^*)→ D^b (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 Rψ_f:D^b(X_η) → D^b (X_s) then sends constructible complexes on X_η to constructible complexes on X_s. 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 Rψ_f 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 Rψ_f 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: