Preparation toward perverse sheaves Alright. Now let us assume that - PowerPoint PPT Presentation

Biduality ◮ Let X be a complex variety. Recall that D ( X ) is the full subcategory of D b (Sh Q ( X )) of constructible sheaves. We claim that D X ◦ D X = id D ( X ) on X and Cheng-Chiang tried to give a proof and failed miserably. ◮ Let me try to sketch the proof I learn from SGA 4 1 2 . ◮ Since H om is local and thus D X is local on X , we may compactify X ; we now assume X is proper. ◮ For any f : X → Y , we have D Y f ∗ = f ∗ D X thanks to that f ∗ = f ! . Suppose we know the result for Y (e.g. by induction and dim Y < dim X ), then we have f ∗ F = D Y D Y f ∗ F = f ∗ D X D X F for any F ∈ D ( X ). ◮ Recall that by filtering degree we may assume F is a shifted sheaf, so for some j : U → X dense open F| U is a shifted local system. ◮ In this case, j ∗ D X D X F = j ∗ F . Hence the mapping cone ∆ of F → D X D X F is supported on Z := X − U . ◮ We saw above that f ∗ ∆ = 0. ◮ Now if Y can be chosen so that the f | Z : Z → Y is finite. Then f ∗ ∆ = 0 ⇒ ∆ = 0 and we are done! Can we?

Preparation toward perverse sheaves ◮ Alright. Now let us assume that D X has all the properties we like. ◮ Recall our first main goal of the series is to define a subcategory Perv( X ) ⊂ D b (Sh Q ( X )) that we claim to have all kinds of good properties. Now we can define: ◮ Definition. p D ≤ 0 ( X ) is the full subcategory consisting of F ∈ D ( X ) such that dim C supp H − k ( F ) ≤ k for all k ∈ Z . In particular H k ( F ) = 0 for k > 0. ◮ Definition. p D ≥ 0 ( X ) is the essential image D X ( p D ≥ 0 ( X )) in D ( X ). ◮ Definition. The category of perverse sheaves is Perv( X ) = “ p D ≤ 0 ( X ) ∩ p D ≥ 0 ( X ) . ” ◮ Example. If X is smooth, then Q X [dim X ] ∈ Perv( X ). For X = V a cone over an elliptic curve in CP 2 in our second problem set, the complex τ ≤− 1 ( j ∗ Q U ) ∈ Perv( V ). ◮ For finite morphisms, f : Z → X , f ∗ = f ! sends Perv( Z ) to Perv( X ). etale morphisms j : U → X , j ∗ = j ! sends Perv( X ) to Perv( U ). For ´

t-structure ◮ To prove properties of Perv( X ) we need some homological algebra preparation, that of t-structure . ◮ To begin with, we had mentioned that Perv( X ) is sort of an upgrade from Sh( X ). The categories analogous to p D ≤ 0 ( X ) and p D ≥ 0 ( X ) are: ◮ Definition. Let D ≤ 0 ( X ) be the full subcategory of D ( X ) with object F such that H k ( F ) = 0 for k > 0. Similarly, D ≥ 0 ( X ) be the full subcategory of D ( X ) with object F such that H k ( F ) = 0 for k < 0. ◮ Write D ≤ n ( X ) := D ≤ 0 ( X )[ − n ] and D ≥ n ( X ) := D ≥ 0 ( X )[ − n ]. We have D ≤ 0 ( X ) ⊂ D ≤ 1 ( X ) and D ≥ 0 ( X ) ⊃ D ≥ 1 ( X ). ◮ For F ∈ D ≤ 0 ( X ) and F ′ ∈ D ≥ 1 ( X ), we have Hom D ( X ) ( F , F ′ ) = 0. ◮ For any F ∈ D ( X ) there exists a distinguished triangle F ′ → F → F ′′ +1 → such that F ′ ∈ D ≤ 0 ( X ) and F ′′ ∈ D ≥ 1 ( X ). − − ◮ Any pair of full subcategories as ( C ≤ 0 , C ≥ 0 ) (as ( D ≤ 0 ( X ) , D ≥ 0 ( X ))) for a triangulated category C (as D ( X )) that satisfies the above three items is called a t-structure . ◮ Of course, the main geometrical claim is that ( p D ≤ 0 ( X ) , p D ≥ 0 ( X )) is a t-structure on the triangulated category D ( X ).

Perverse t-structure ◮ For any t-structure ( C ≤ 0 , C ≥ 0 ) of a triangulated category C , the core of the t-structure is defined to be the full subcategory C 0 of objects that are (isomorphic to objects) in both C ≤ 0 and C ≥ 0 . ◮ Theorem. (i) The core of a t-structure is an abelian category. ◮ (ii) The natural functors C ≤ 0 → C admits a right adjoint τ ≤ 0 : C → C ≤ 0 . Dually, C ≥ 0 → C admits a left adjoint τ ≥ 0 : C → C ≥ 0 . ◮ (iii) There is a natural isomorphism of functors τ ≥ 0 ◦ τ ≤ 0 = τ ≤ 0 ◦ τ ≥ 0 . ◮ (iv) Let us temporarily denote τ ≥ 0 ◦ τ ≤ 0 by H 0 ( − ) and likewise H k ( − ) := H 0 ( − [ k ]). For any distinguished triangle F ′ → F → F ′′ +1 − − → , we have the long exact sequence ... → H k ( F ′ ) → H k ( F ) → H k ( F ′′ ) → H k +1 ( F ′ ) → ... ◮ (v) For F ∈ C , we have F ∈ C ≤ 0 iff H k ( F ) = 0 for k > 0. Dually F ∈ C ≥ 0 iff H k ( F ) = 0 for k < 0.

Perverse t-structure, II ◮ (vi) Let F ′ ∈ C ≤ 0 and F ′′ ∈ C ≥ 0 . Then any morphism F ′ → F ′′ factors through F ′ → H 0 ( F ′ ) → H 0 ( F ′′ ) → F ′′ . In fact we have a ∼ natural isomorphism Hom C 0 ( H 0 ( F ′ ) , H 0 ( F ′′ )) → Hom C ( F ′ , F ′′ ). − ◮ (vii) A sequence 0 → F ′ → F → F ′′ → 0 with F , F ′ , F ′′ ∈ C 0 is short exact iff F ′ → F → F ′′ can be completed to a distinguished triangle in C . ◮ Oh great! So now we take C = D ( X ), C ≤ 0 = p D ≤ 0 ( X ), C ≥ 0 = p D ≥ 0 ( X ) and C 0 = Perv( X ). All the good properties hold, as long as we prove that this is really a t-structure. ◮ We have to prove that 1. For any F ′ , F ′′ with dim supp H − k ( F ′ ) ≤ k and dim supp H − k − 1 ( F ′′ ) ≤ k , so that F ′ ∈ p D ≤ 0 ( X ) and F ′′ ∈ p D ≤− 1 ( X ), we have Hom D ( X ) ( F ′ , D X F ′′ ) = 0. 2. For any F ∈ D ( X ), we can fit F into a distinguished triangle F ′ → F → D X F ′′ such that F ′ and F ′′ are as above. ◮ Scary enough. Allow me to continue next time.