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

• f Db(ShQ(X)) of constructible sheaves. We claim that

DX ◦ DX = idD(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 Hom is local and thus DX is local on X, we may compactify X; we now assume X is proper. ◮ For any f : X → Y , we have DY f∗ = f∗DX 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 = DY DY f∗F = f∗DXDXF 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∗DXDXF = j∗F. Hence the mapping cone ∆ of F → DXDXF 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 DX has all the properties we like. ◮ Recall our first main goal of the series is to define a subcategory Perv(X) ⊂ Db(ShQ(X)) that we claim to have all kinds of good

• properties. Now we can define:

◮ Definition. pD≤0(X) is the full subcategory consisting of F ∈ D(X) such that dimC supp H−k(F) ≤ k for all k ∈ Z. In particular Hk(F) = 0 for k > 0. ◮ Definition. pD≥0(X) is the essential image DX(pD≥0(X)) in D(X). ◮ Definition. The category of perverse sheaves is Perv(X) = “pD≤0(X) ∩ pD≥0(X).” ◮ Example. If X is smooth, then QX[dim X] ∈ Perv(X). For X = V a cone over an elliptic curve in CP2 in our second problem set, the complex τ≤−1(j∗QU) ∈ Perv(V ). ◮ For finite morphisms, f : Z → X, f∗ = f! sends Perv(Z) to Perv(X). For ´ etale morphisms j : U → X, j∗ = j! sends Perv(X) to Perv(U).

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 pD≤0(X) and pD≥0(X) are: ◮ Definition. Let D≤0(X) be the full subcategory of D(X) with object F such that Hk(F) = 0 for k > 0. Similarly, D≥0(X) be the full subcategory of D(X) with object F such that Hk(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 HomD(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 (pD≤0(X), pD≥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

• f the t-structure is defined to be the full subcategory C0 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 H0(−) and likewise Hk(−) := H0(−[k]). For any distinguished triangle F′ → F → F′′

+1

− − →, we have the long exact sequence ... → Hk(F′) → Hk(F) → Hk(F′′) → Hk+1(F′) → ... ◮ (v) For F ∈ C, we have F ∈ C≤0 iff Hk(F) = 0 for k > 0. Dually F ∈ C≥0 iff Hk(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′ → H0(F′) → H0(F′′) → F′′. In fact we have a natural isomorphism HomC0(H0(F′), H0(F′′))

∼

− → HomC(F′, F′′). ◮ (vii) A sequence 0 → F′ → F → F′′ → 0 with F, F′, F′′ ∈ C0 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 = pD≤0(X), C≥0 = pD≥0(X) and C0 = 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 ′ ∈ pD≤0(X) and F ′′ ∈ pD≤−1(X), we have HomD(X)(F′, DXF ′′) = 0.

• 2. For any F ∈ D(X), we can fit F into a distinguished triangle

F ′ → F → DXF ′′ such that F ′ and F ′′ are as above.

◮ Scary enough. Allow me to continue next time.