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 (over Q ). A complex is called constructible if its cohomology are. ◮ We denote by D ( X ) the full subcategory of D b (Sh Q ( X )) of constructible complexes. ◮ Lemma. R H om and ⊗ are well-defined in D ( X ). ◮ Theorem. If f : X → Y is a morphism of varieties. Then f ∗ , Rf ∗ , Rf ! , f ! sends constructible complexes to constructible complexes. ◮ Recall D X := R H om X ( − , ( X → pt ) ! Q ). Now the big result is: ◮ Theorem. Restricting to D ( X ) we have D X ◦ D X = id D ( X ) . ◮ Before we discuss the proofs, let’s see the immediate applications. ◮ Working with D ( X ) , we will abbreviate Rf ∗ to f ∗ , Rf ! to f ! , and R H om to H om from now on.

Properties of D X D X := H om X ( − , ( X → pt ) ! Q ). Restricting to D ( X ) we have D X ◦ D X = id D ( X ) . ◮ Theorem implies that D X is an (anti-)equivalence D X : D ( X ) → D ( X ) op . ∼ ◮ That is, Hom D ( X ) ( F , F ′ ) → Hom D ( X ) ( D X F ′ , D X F ). − ◮ In Verdier duality for f : X → Y a morphism of complex varieties: H om Y ( f ! F , G ) = f ∗ H om X ( F , f ! G ) ◮ Plugging in G = ω Y := ( Y → pt ) ! Q (the “dualizing complex”), we get H om Y ( f ! F , ω Y ) = f ∗ H om X ( F , ω X ) which is D Y ◦ f ! = f ∗ ◦ D X . ◮ Applying D Y on the left and D X on the right, we get f ! ◦ D X = D Y ◦ f ∗ .

Duality revisited D Y ◦ f ! = f ∗ ◦ D X , f ! ◦ D X = D Y ◦ f ∗ . ◮ Adjunction gives functorial isomorphisms Hom D ( X ) ( F , f ! D Y G ) = Hom D ( Y ) ( f ! F , D Y G ) = Hom D ( Y ) ( G , D Y f ! F ) = Hom D ( Y ) ( G , f ∗ D X F ) = Hom D ( X ) ( f ∗ G , D X F ) = Hom D ( X ) ( F , D X f ∗ G ) . ◮ By Yoneda lemma we have f ! ◦ D Y = D X ◦ f ∗ . ◮ Dualizing again we have f ∗ ◦ D Y = D X ◦ f ! . ◮ Now, let us try to prove the assertions that all our functors preserve D ( − ) and D X ◦ D X = id on D ( X ).

Some distinguished triangles ◮ Let us first note the very important fact that D ( X ) is a triangulated subcategory of D b (Sh( X )): this is essentially that in a distinguished triangle F → F ′ → F ′′ +1 → in D b (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 +1 put F into τ ≤ n F → F → τ > n F − − → . 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 +1 j ! j ! F → F → i ∗ i ∗ F − − → ◮ It works for any F ∈ D (Sh( X )), but we have F ∈ D ( X ) iff j ! j ! F and i ∗ i ∗ F do.

Distinguished triangles ◮ Let us first note the very important fact that D ( X ) is a triangulated subcategory of D b (Sh( X )): this is essentially that in a distinguished triangle F → F ′ → F ′′ +1 → in D b (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 +1 put F into τ > n F → F → τ ≤ n F − − → . 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 +1 j ! j ! F → F → i ∗ i ∗ F − − → ◮ It works for any F ∈ D (Sh( X )), but we have F ∈ D ( X ) iff j ! j ! F and i ∗ i ∗ F do.

Stratification ◮ Alright! So now we can decompose F ∈ D ( X ) until it is just (a shift of) 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 H k c ( f ∗ F ) where H k ( 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 D X ◦ D X = id on D ( X ), note that for any F ∈ D ( X ) we can look at +1 j ! j ! F i ∗ i ∗ F F +1 D X D X j ! j ! F D X D X i ∗ i ∗ F D X D X F ◮ And it suffice to prove that j ! j ! F ∼ = D X D X j ! j ! F and i ∗ i ∗ F ∼ = D X D X i ∗ i ∗ F naturally.

Open and closed It suffice to prove that D X D X j ! j ! F = j ! j ! F and D X D X i ∗ i ∗ F = i ∗ i ∗ F naturally. ◮ In fact we claim that D X D X i ∗ G = i ∗ G naturally for any G ∈ D ( Z ). This follows from induction as D X i ∗ = i ∗ D Z ; our proof of this didn’t use the properties of D X . It can also be proved by hand. ◮ Similarly, we claim that D X D X j ! 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 D X j ! G = H om X ( j ! G , ω X ), we consider ω X as in +1 j ! ω U → ω X → i ∗ i ∗ ω X − − → ◮ We have H om X ( j ! G , i ∗ i ∗ ω X ) = j ∗ H om U ( G , j ! i ∗ i ∗ ω X ) = 0. Hence H om X ( j ! G , ω X ) = H om X ( j ! G , j ! ω U ). ◮ Finally we have D X D X j ! G = D X ( H om X ( j ! G , j ! ω U )) = H om X ( H om X ( j ! G , j ! ω U ) , ω X ) = H om X ( j ! ω U , ω X ⊗ j ! G ) = H om ( Q U [dim X ] , j ! G [dim X ]) = j ! G .