SLIDE 1
Properties of DX
DX := Hom X (−, (X → pt)!Q). Restricting to D(X) we have DX ◦ DX = idD(X).
◮ Theorem implies that DX is an (anti-)equivalence DX : D(X) → D(X)op. ◮ That is, HomD(X)(F, F′)
∼
Duality revisited D Y f ! = f D X , f ! D X = D Y f . - - PowerPoint PPT Presentation
Duality revisited D Y f ! = f D X , f ! D X = D Y f . - - PowerPoint PPT Presentation
Constructible sheaves Definition. Let X be a variety over C . Say F Sh Q ( X ) is constructible if we have a stratification X = X 0 X 1 ... by closed subvarieties such that F| X i X i +1 is a locally constant sheaf of finite rank
SLIDE 2
SLIDE 3
Duality revisited
DY ◦ f! = f∗ ◦ DX , f! ◦ DX = DY ◦ f∗.
◮ Adjunction gives functorial isomorphisms HomD(X)(F, f !DY G) = HomD(Y )(f!F, DY G) = HomD(Y )(G, DY f!F) = HomD(Y )(G, f∗DXF) = HomD(X)(f ∗G, DXF) = HomD(X)(F, DXf ∗G). ◮ By Yoneda lemma we have f ! ◦ DY = DX ◦ f ∗. ◮ Dualizing again we have f ∗ ◦ DY = DX ◦ f !. ◮ Now, let us try to prove the assertions that all our functors preserve D(−) and DX ◦ DX = id on D(X).
SLIDE 4
Some distinguished triangles
◮ Let us first note the very important fact that D(X) is a triangulated subcategory of Db(Sh(X)): this is essentially that in a distinguished triangle F → F′ → F′′
+1
− − → in Db(Sh(X)), if any two complex is constructible, then so is the third thanks to long exact sequence of the cohomology. ◮ If we want to prove e.g. f∗F is constructible assuming F is, we may put F into τ≤nF → F → τ>nF
+1
− − →. By filtering it repeatedly, we can reduce to actual constructible sheaves. ◮ Let i : Z ֒ → X be a closed subvariety and j : U ֒ → X be the complementary subvariety. We have the most important distinguished triangle j!j!F → F → i∗i∗F
+1
− − → ◮ It works for any F ∈ D(Sh(X)), but we have F ∈ D(X) iff j!j!F and i∗i∗F do.
SLIDE 5
Distinguished triangles
◮ Let us first note the very important fact that D(X) is a triangulated subcategory of Db(Sh(X)): this is essentially that in a distinguished triangle F → F′ → F′′
+1
− − → in Db(Sh(X)), if any two complex is constructible, then so is the third thanks to long exact sequence of the cohomology. ◮ If we want to prove e.g. f∗F is constructible assuming F is, we may put F into τ>nF → F → τ≤nF
+1
− − →. By filtering it repeatedly, we can reduce to actual constructible sheaves. ◮ Let i : Z ֒ → X be a closed subvariety and j : U ֒ → X be the complementary subvariety. We have the most important distinguished triangle j!j!F → F → i∗i∗F
+1
− − → ◮ It works for any F ∈ D(Sh(X)), but we have F ∈ D(X) iff j!j!F and i∗i∗F do.
SLIDE 6
Stratification
◮ Alright! So now we can decompose F ∈ D(X) until it is just (a shift
- f) a locally constant sheaf (or local system) on a locally closed
subvariety. ◮ If we want to compute f!F, we can further decompose it, so that f restricted to the locally closed subvariety is a fibration. It then follows that f!F is a direct sum of Hk
c (f∗F) where Hk(f∗F) is the
k-th compactly supported cohomology of the fiber(s). ◮ Other functors are more difficult. Meanwhile, assuming it’s good for all other functors. To show that DX ◦ DX = id on D(X), note that for any F ∈ D(X) we can look at j!j!F F i∗i∗F DXDXj!j!F DXDXF DXDXi∗i∗F
+1 +1
◮ And it suffice to prove that j!j!F ∼ = DXDXj!j!F and i∗i∗F ∼ = DXDXi∗i∗F naturally.
SLIDE 7
Open and closed
It suffice to prove that DX DX j!j!F = j!j!F and DX DX i∗i∗F = i∗i∗F naturally.
◮ In fact we claim that DXDXi∗G = i∗G naturally for any G ∈ D(Z). This follows from induction as DXi∗ = i∗DZ; our proof of this didn’t use the properties of DX. It can also be proved by hand. ◮ Similarly, we claim that DXDXj!G = j!G where G = j!F may be assumed to be just a local system. We may also assume U is smooth. ◮ To compute DXj!G = Hom X(j!G, ωX), we consider ωX as in j!ωU → ωX → i∗i∗ωX
+1