Poincar e-Verdier duality Having proved Verdier duality, our next - - PowerPoint PPT Presentation

Verdier duality

◮ Last time, we stated for f : X → Y map between “finite-dimensional,” locally compact Hausdorff spaces: RHomSh(Y )(Rf!F, G) = RHomSh(X)(F, f !G) ∈ D+(Ab). R Hom Y (Rf!F, G) = Rf∗R Hom X(F, f !G) ∈ D+(Sh(Y )). ◮ The second does imply the first, as RΓ ◦ R Hom = RHom; this is because Hom (F, G) is flabby whenever G is injective. ◮ Let us make some preparation. For any F, F′ ∈ Sh(X), we have a natural morphism f∗Hom X(F, F′) → Hom Y (f∗F, f∗F′) ◮ Since sections with compact supports are mapped to sections with compact support, this induces f∗Hom X(F, F′) → Hom Y (f!F, f!F′) ◮ Now we want to derive both to Rf∗R Hom X(F, F′) → R Hom Y (Rf∗F, Rf∗F′) Rf∗R Hom X(F, F′) → R Hom Y (Rf!F, Rf!F′)

Sheaf Hom and derived functors

f∗Hom X (F, F′) → Hom Y (f∗F, f∗F′), f∗Hom X (F, F′) → Hom Y (f!F, f!F′)

◮ There are some issues. To begin with, to have Rf∗F or Rf!F on the left of a derived Hom we need the complex to be in D−(Sh(X)). Even if we assume F ∈ Db(Sh(X)) (or just a sheaf), we still need Rf∗ to preserve the bounded above property. ◮ There is a fact that flabby dimension ≤ c-soft dimension +1. That is, if we know Hi

c(X; F) = 0 for F ∈ Sh(X) and all i > n, then we

can conclude every sheaf F has a flabby resolution 0 → F → I0 → I1 → ... → In+1 → 0. ◮ Using this, we have F ∈ Db(Sh(X)) ⇒ Rf∗F ∈ Db(Sh(X)). We already know this for Rf!F. ◮ Now how do we derive the transformations at the top? For the first, we can use a flabby resolution of F′ given by direct product of skyscraper sheaves. Then f∗F′ is also a direct product of skyscraper sheaves, and so is Hom X(F, F′), so we are happy. ◮ But what about the f! one? I think this should be do-able by replacing flabby with c-soft appropriately, but I haven’t figured out

• how. Kashiwara-Shapira leaves this as an exercise (2.6.25).

Sheaf version Verdier duality

Rf∗R Hom X (F, F′) → R Hom Y (Rf!F, Rf!F′)

◮ Now we have Rf∗R Hom X(F, f !G) → R Hom (Rf!F, Rf!f !G) → R Hom (Rf!F, G) thanks to the ordinary Verdier duality Rf! ◦ f ! → idD+(Sh(Y )). To prove that this an (quasi)-isomorphism, we verify for any U ⊂ Y

• pen

Rkf∗R Hom X(F, f !G)(U) = Hk(Rf∗R Hom X(F, f !G)(U)) = Hk(RΓ(U, Rf∗R Hom (F, f !G))) = Hk(RΓ(f −1(U), R Hom (F, f !G))) = Hk(RHomSh(f −1(U))(F, f !G)) = HomSh(f −1(U))(F, f !G[k]) = HomSh(U)(Rf!F, G[k]) = Hk(RHomSh(U)(Rf!F, G)) = Hk(RΓ(U, R Hom U(Rf!F, G))) = RkR Hom Y (Rf!F, G)(U) Here we have abuse F for F|f −1(U), etc. The sixth equality - the one at the beginning of fourth line - is the original Verdier duality.

Poincar´ e-Verdier duality

◮ Having proved Verdier duality, our next goal is to compute f !G, at least for very nice maps for constant sheaf? The most important assertion is that: ◮ Theorem. Suppose f : X → Y is a topological submersion of (real) dimension n (i.e. locally on U ⊂ X it looks like U = f (U) × Rn → f (U)) where Y is locally compact Hausdorff. Then f !ZY = OY /X[n], where OY /X is the locally constant sheaf on X with fiber Z that records the orientation of the fibers. ◮ Corollary. Suppose f : X → Y is a smooth morphism of complex variety of (complex) dimension n. Then f !ZY = ZX[2n]. ◮ How do we prove this? The expectation that f !ZY concentrates at

• ne degree helps, cause it suffices to compute Hk(f !ZY ).

◮ But Hk(f !ZY ) is the sheafification of U → RkΓ(U; f !Z|U). ◮ We have RΓ(U; (f !ZY )|U) = RHomSh(U)(ZU, (f !ZY )|U) = RHomSh(f (U))(Rf!ZU, Zf (U)).

Poincar´ e-Verdier duality, II

Hk(f !ZY ) is the sheafification of U → RkΓ(U; f !Z|U), and RkΓ(U; f !Z|U) = Hk RHomSh(f (U))(Rf!ZU, Zf (U))

◮ It suffices to consider those U shrunk so that U = f (U) × Rn. Proper base change tells us Rkf!ZU is locally (on f (U)) given by Hk

c (Rn; Z) =

0, if k = n Z, if k = n ◮ We have RHomSh(f (U))(Rf!ZU, Zf (U)) = RHomSh(f (U))(Z[−n], Z). ◮ If we further shrink U so that U is contractible1, then RHomSh(f (U))(Z[−n], Z) = Z. ◮ This proves that f !ZY only supported at degree −n with a locally constant sheaf with stalk Z there. Carefully tracking the isomorphism Hn

c (Rn; Z) ∼

= Z will lead us to the fiber/relative

• rientation sheaf - let’s leave it to next time.

1This assumes Y locally contractible; see [KS, Prop. 3.3.2] for a more careful proof

without this assumption.