Simple objects on Z Fix any i : Z X closed and complementary j : U - - PowerPoint PPT Presentation

Simple objects on Z

◮ Fix any i : Z ֒ → X closed and complementary j : U → X open for

• today. At the end of last time we say we would like

“0 → Perv(Z)

i∗=i!

− − − → Perv(X)

j∗=j!

− − − → Perv(U) → 0′′ ◮ In fact, I think there are good reason that the above won’t be true with appropriate interpretation (See next problem set) ◮ The above is true on the level of pointed set (or groupoid?). Indeed, if j∗F = 0 for F ∈ Perv(X), then F = i∗i∗F is in the essential image i∗(Perv(Z)). ◮ We furthermore note that i∗ = i! : Perv(Z) → Perv(X) is fully faithful, for pH0 ◦ i∗ is the inverse functor from the essential image back to Perv(Z). ◮ Moreover, suppose G is a simple object in Perv(Z), we claim that i∗G is simple.

Simple objects on U

Suppose G is a simple object in Perv(Z), we claim that i∗G is simple.

◮ One important thing I failed to stress last time: a t-exact (resp. left, right exact) functor of two triangulated category equipped with their

• wn t-structures induces exact (resp. left, right exact) functor on

the core/heart. ◮ Now for G ∈ Perv(Z) simple, if 0 → F′ → i∗G → F′′ → 0 in Perv(X) we would have 0 → j∗F′ → 0 → j∗F′′ → 0 so j∗F′ = j∗F′′ = 0, i.e. F′, F′′ are in the essential image of Perv(Z). But then fully faithfulness shows F′ = 0 or F′′ = 0. This proves that i∗G is simple. ◮ On the other hand, let us assume that F ∈ Perv(X) is simple, and j∗F = 0. We claim that: ◮ Theorem. The set of (isomorphism classes of) such simple F is in bijection via j∗ with simple objects Perv(U). ◮ What do we do? So the most interesting question is probably: given simple G ∈ Perv(U), how do we find a simple F with F|U = G? ◮ Naive approach will be either j!G or j∗G. We have j!G ∈ pD≤0(X) and j∗G ∈ pD≥0(X) and not perverse in general.

Middle extension

Naive approach will be either j!G or j∗G. We have j!G ∈ pD≤0(X) and j∗G ∈ pD≥0(X) and not perverse in general.

◮ A simple fix: one may take either pτ≥0j!G = pH0(j!G) and/or

pτ≤0j∗G = pH0(j∗G). However as we will see in the problem session,

they are usually not simple when G is. ◮ What do we do? Let’s try to fix any of them. Say we look at any, e.g. pτ≤0j∗G. ◮ How did we construct it last time? Since j∗G has the correct perverse degree on U (i.e. correct degree with respect to the perverse t-structure), the trouble happens on Z, and we define

pτ≤0j∗G as the objects that fits into pτ≤0j∗G → j∗G → i∗ pτ≥1i∗j∗G +1

− − → ◮ Now, if we want something whose pH0 “only comes from G,” it seems like we should get rid of the perverse degree 0 part from Z, suggesting to look at the object K that fits into K → j∗G → i∗

pτ≥0i∗j∗G +1

− − →

Middle extension, II

K → j∗G → i∗pτ≥0i∗j∗G

+1

− − →.

◮ Proposition. Fix G ∈ Perv(U). Suppose K is an object in D(X) with given isomorphism j∗K ∼ = G. TFAE:

(i) K ∈ Perv(X) is the object fitting the distinguished triangle above. (ii) i∗K ∈ pτ≤−1(Z) and i!K ∈ pτ≥1(Z). (iii) K ∈ Perv(X) does not have any non-trivial sub-object nor quotient

• bject from Perv(Z).

(iv) K is the image in Perv(X) of the natural map pH0(j!G) → pH0(j∗G).

◮ Let us prove (ii) implies (i). From j∗K ∼ = G we have by adjunction K → j∗G = j∗j∗K which fits into a distinguished triangle i∗i!K → K → j∗j∗K

+1

− − →. We would like to prove that j∗G = j∗j∗K → i∗i!K[1] is equal to j∗G → i∗pτ≥0i∗j∗G. ◮ By adjunction, we have to prove that the second map in the triangle i∗K → i∗j∗G → i!K[1]

+1

− − → is equal to i∗j∗G → pτ≥0i∗j∗G. ◮ Such a statement is equivalent to i∗K ∈ pD≤−1(Z) and i!K[1] ∈ pD≥0(Z). This is exactly (ii).

Middle extension, III

(i) K ∈ Perv(X) is the object fitting K → j∗G → i∗pτ≥0i∗j∗G

+1

− − →. (ii) i∗K ∈ pτ≤−1(Z) and i!K ∈ pτ≥1(Z). (iii) K ∈ Perv(X) has no non-trivial sub/quotient object from Perv(Z). (iv) K is the image in Perv(X) of the natural map pH0(j!G) → pH0(j∗G).

◮ Next prove (iii)⇒(ii). It is known that i∗K ∈ pD≤0(Z). Suppose (iii) holds but on the contrary i∗K ∈ pD≤−1(Z). Then adjunction gives a non-trivial morphism K → i∗pH0(i∗K) in Perv(X) whose image is a quotient object of K supported on Z. This contradiction and its dual version proves (iii) = ⇒ (ii). ◮ Now we prove (iv)⇒(iii). Suppose (iv) but we have an epimorphism K → i∗A with non-zero A ∈ Perv(Z). We have j!G → pH0(j!G) → K → i∗A. The whole composition is zero as it’s adjoint to G → j!i∗A = 0. Since i∗A ∈ pD≥0(X), this implies that the composition pH0(j!G) → K → i∗A is also zero. But this contradicts with that pH0(j!G) → K is surjective. ◮ So we have (iv)⇒(iii)⇒(ii)⇒(i). But both (i) and (iv) uniquely characterize K and thus (i)⇔(iv) and we are done.

Middle extension, IV

(i) K ∈ Perv(X) is the object fitting K → j∗G → i∗pτ≥0i∗j∗G

+1

− − →. (ii) i∗K ∈ pτ≤−1(Z) and i!K ∈ pτ≥1(Z). (iii) K ∈ Perv(X) has no non-trivial sub/quotient object from Perv(Z). (iv) K is the image in Perv(X) of the natural map pH0(j!G) → pH0(j∗G).

◮ We have shown that for G ∈ Perv(U), there exists a unique extension K ∈ Perv(X) satisfying the nice properties. Since properties (i) and (iv) are functorial in G, this defines a functor j!∗ : Perv(U) → Perv(X) for an Zariski open j : U ֒ → X. The object K = j!∗G is typically called the IC/intermediate/middle extension

• f G.

◮ We have that G is simple iff j!∗G. The only if statement follows from applying the t-exact functor j∗ = j!. For the if statement, suppose j!∗G was not simple, by (iii) it has 0 → F′ → j!∗G → F′′ → 0 in Perv(X) with F′, F′′ not supported on Z, i.e. j∗F′, j∗F′′ non-zero. ◮ Applying the t-exact functor j∗ then gives 0 → j∗F′ → G → j∗F′′ → 0, a contradiction.

Middle extension, V

◮ We have shown that for G ∈ Perv(U), there exists a unique extension K ∈ Perv(X) satisfying properties (i)-(iv). Since (i) and (iv) are functorial in G, this defines a functor j!∗ : Perv(U) → Perv(X) for an Zariski open j : U ֒ → X. The object K = j!∗G is typically called the IC/intermediate/middle extension of G. ◮ We have by property (ii), (iii), (iv) above that j!∗ ◦ DU = DX ◦ j!∗. ◮ Lemma. j!∗ is left and right exact (but not middle exact, see problem set 3). ◮ Proof. We prove that j!∗ is left exact. The right exact statement follows either applying DX to the argument, or to the exact

• sequence. Now suppose we have G′ → G a monomorphism in

Perv(U) and we would like to show that j!∗G′ → j!∗G is a

• monomorphism. Suppose we have F → j!∗G′ → j!∗G so that the

composition is zero. We have to show F → j!∗G′ is zero. ◮ Applying j∗ gives j∗F → G zero. Since G′ → G is a monomorphism, we have j∗F → G′ zero too. By adjunction this means the composition F → j!∗G′ → j∗G is zero. But j!∗G′ → j∗G is a monomorphism then implies F → j!∗G′ is zero, as desired.

Simple objects

◮ Lemma. G ∈ Perv(U) is simple iff j!∗G ∈ Perv(X) is. ◮ For the only if statement, if j!∗G was not simple, by (iii) it has sub and quotient object not supported on Z. But then j∗ again shows G is not simple. ◮ For the if statement, suppose otherwise G has a proper sub-object A ֒ → G. Then we have j!∗A → j!∗G. This map is both non-trivial and not surjective as both can be detected by j∗ = j!. Hence j!∗G is not simple.

Simple objects, II

◮ Suppose now X is irreducible smooth and L is a local system. Then L[dim X] ∈ Perv(X). ◮ Lemma. L is simple as a local system, i.e. an object of Rep(π1(X)) iff L[dim U] is simple in Perv(X). ◮ A short exact sequence 0 → L′ → L → L′′ → 0 in Rep(π1(X)) induces a distinguished triangle in D(X) and thus another short exact sequence 0 → L′[dim X] → L[dim X] → L′′[dim X] → 0 in Perv(X). This proves the if part. ◮ For the only if part, we note that L[dim X] satisfies the property (ii) in the criterion of middle extension; i∗L[dim X] ∈ pD≤−1(Z) for any i : Z → X with dim Z < dim X. Hence if L[dim X] has a proper sub-object E in Perv(X), it’s restriction to any open set must be non-trivial. ◮ We may then find a small enough Zariski open U so that j∗E = L′[dim X] for some L′ ∈ Rep(π1(U)) and we have L′ ֒ → L|U. But this contradicts with the simplicity of L as π1(U) → π1(X) is surjective; any loop can avoid real codimension two strata.

Simple objects, III

◮ Let us give a summary: any simple perverse sheaf on a general X comes from a closed subvariety Z. It is then the middle extension of a simple perverse sheaf from any dense open V . ◮ By shrinking the dense open we may assume V is smooth and the perverse sheaf is just a shifted local system. Now V has to be connected, hence irreducible. This gives: ◮ Corollary. For general X, any simple object in Perv(X) is of the form i∗j!∗L[dim V ] for some i : Z ֒ → X irreducible closed, j : V ֒ → Z dense open smooth, and L a simple local system on V . Any object

• f the form is indeed simple. The choice of Z is unique, and

different choices of (V , L) agrees on their intersection. ◮ This object is typically written IC(Z; L). One also writes ICZ = IC(Z) := IC(Z; QV ). ◮ For Z = X one also write IH∗(X; L) := RΓ(X; IC(X; L)), the intersection cohomology of F on X. In particular IH∗(X) := IH∗(X; QV ) = RΓ(ICX).

Remark

◮ Lemma. The functor j!∗ : Perv(U) → Perv(X) is fully faithful. ◮ Proof. Since j∗ ◦ j!∗ = idPerv(U), we have HomPerv(U)(G, G′) ֒ → HomPerv(X)(j!∗G, j!∗CG ′). We want to prove this is surjective. ◮ We have HomPerv(U)(G, G′) = HomD(U)(j∗j!∗G, G′) = HomD(X)(j!∗G, j∗G′). ◮ Since j∗G′ ∈ pD≥0(X), we have HomD(X)(j!∗G, j∗G′) = HomPerv(X)(j!∗G, pH0(j∗G′)). ◮ Recall that we have 0 → j!∗G′ → pH0(j∗G′) → i∗A → 0 in Perv(X) for some A ∈ Perv(Z). Applying HomPerv(X)(j!∗G, −) gives 0 → HomPerv(X)(j!∗G, j!∗G′) → HomPerv(X)(j!∗G, pH0(j∗G′)) → HomPerv(X)(j!∗G, i∗A) Since the last term is zero as j!∗G has no quotient object on Z, the lemma is proved.

Computing j!∗

◮ Alright! So the most fun question is: how do we compute j!∗, IC(−) and IH∗(−)? ◮ Before doing it next time, let us look at the example where we have V is the affine cone over an elliptic curve with base point o, and j : U := V − {o} ֒ → V . We would like to find ICV = j!∗QU[2]. ◮ By (i) of our defining proposition (here i : {o} ֒ → V ): ICV → j∗QU[2] → i∗pτ≥0i∗j∗QU[2]

+1

− − →. ◮ But pτ≥0 on D({o}) is just τ≥0! One sees that i∗pτ≥0i∗j∗QU[2] = τ≥0j∗QU[2] and ICV = τ≤−1j∗QU[2]. ◮ In general, if we have X = Xn ⊃ Xn−1 ⊃ ... ⊃ X0 ⊃ X−1 = ∅ with each Xi closed and G ∈ Perv(Xn − Xn−1). Define ji : Xn − Xi ֒ → Xn − Xi−1 for i = n − 1, n − 2, ..., 0. Then under certain smoothness conditions we expect IC(X, G) = τ≤−1(j0)∗τ≤−2(j1)∗...τ≤−n+1(jn−2)∗τ≤−n(jn−1)∗G. Let us go over this carefully next time.

Deligne’s formula for j!∗

◮ Suppose we have X = Xn ⊃ Xn−1 ⊃ ... ⊃ X0 ⊃ X−1 = ∅ with each Xa Zariski closed. Write Ua := Xn − Xa for a = n − 1, ..., 0, −1. We assume Xa − Xa−1 = Ua−1 − Ua is smooth of dimension a. Fix L ∈ Loc(Un−1) (Loc(−) for the abelian category of finite rank Q-local systems) so Gn−1 := L[n] ∈ Perv(Un−1). Write ja : Ua ֒ → Ua−1. Define inductively Ga−1 := τ≤−a−1(ja)∗Ga, a = n − 1, n − 2, ..., 0. ◮ Theorem. Suppose for each a we have Hk((ja)∗Ga)|Xa−Xa−1 are local systems for each k, then each Ga = IC(Ua; L). In particular G−1 = IC(X; L). ◮ Proof. It suffices to do it for one step; assume that Ga = IC(Ua; L) we have to prove Ga−1 = IC(Ua−1; L). By transitivity of j!∗, we know IC(Ua−1; L) = (ja)!∗Gi and we have to prove (ja)!∗Ga ∼ = τ≤−a−1(ja)∗Ga. ◮ Equivalently, we have to prove that (ja)!∗Ga → (ja)∗Ga → τ≥−a(ja)∗Ga

+1

− − → is a distinguished triangle. ◮ By one definition of j!∗, it suffices to prove τ≥−a(ja)∗Ga = i∗pτ≥0i∗(ja)∗Ga.

Deligne’s formula for j!∗, II

It suffices to prove τ≥−a(ja)∗Ga = i∗pτ≥0i∗(ja)∗Ga.

◮ Since Ga = τ≤−a−1Ga by construction, we have for i : Ua−1 − Ua = Xa − Xa−1 ֒ → X − Xa−1 = Ua−1 that τ≥−a(ja)∗Ga = i∗τ≥−ai∗(ja)∗Ga. Hence it remains to prove that τ≥−ai∗(ja)∗Ga = pτ≥0i∗(ja)∗Ga as two complexes of sheaves on Ua−1 − Ua. The theorem thus follows from the following lemma: ◮ Lemma. Let Y be an n-(equi-)dimensional smooth variety and F ∈ D(Y ) such that Hk(F) are local systems for each k. Then

pτ≤k(F) = τ≤k−dim Y (F) and pτ≥k(F) = τ≥k−dim Y (F).

◮ The Lemma will follow - in fact equivalent - to that τ≤k−dim Y (F) ∈ pD≤k(F) and τ≥k−dim Y (F) ∈ pD≥k(F). ◮ To prove the last statement, filter F with truncation τ≤m(F) → F → τ≥m+1(F)

+1

− − → so that F is supported at one degree, and hence a shifted local system. The result then follows from definition and that DY L = L[2 dim Y ] for L ∈ Loc(Y ).

Deligne’s formula for j!∗, II

It suffices to prove τ≥−a(ja)∗Ga = i∗pτ≥0i∗(ja)∗Ga.

◮ Since Ga = τ≥−aGa by construction, it remains to prove that

pτ≥0i∗(ja)∗Ga = τ≥−ai∗(ja)∗Ga as two complexes of sheaves on

Ua−1 − Ua. The theorem thus follows from the following lemma: ◮ Lemma. Let Y be an n-(equi-)dimensional smooth variety and F ∈ D(Y ) such that Hk(F) are local systems for each k. Then

pτ≤k(F) = τ≤k−dim Y (F) and pτ≥k(F) = τ≥k−dim Y (F).

◮ By filtering F with truncation τ≤m(F) → F → τ≥m+1(F)

+1

− − → it suffices to show this when F is supported at one degree, and hence a shifted local system. The result then follows from definition and that DY L = L[2d Im Y ] for L ∈ Loc(Y ). ◮ Cheng-Chiang’s favorite case; algebraic group G acts on X with an

• pen dense orbit, so Un−1 is that orbit, and Ua are unions of a finite

number of smaller orbits. If the sheaves comes from something with G-action (eventually equivariant sheaves), then the hypothesis of Deligne’s formula is satisfied.

Perv(X) is artinian and noetherian

◮ Prop. Every object in Perv(X) is of finite length, i.e. the category is noetherian and artinian. ◮ Let F ∈ Perv(X). Take U ⊂ X Zariski open such that dim Z < dim X (Z := X − U) and that Hk(F|U) are local systems, which by perversity implies F|U = L[dim X] for some L ∈ Loc(U). ◮ We have i∗i∗F → F → j!L[dim X]

+1

− − →, which induces the long exact sequence in perverse cohomology ... → pH0(i∗i∗F) → pH0(F) → pH0(j!L[dim X]) → ... Note pH0(i∗i∗F) = i∗pH0(i∗F) and pH0(F) = F. We have seen that sub/quotient objects of objects from Perv(Z) (via i∗) are still in Perv(Z). Hence pH0(i∗i∗F) is of finite-length.

“Cohomology” only lives in positive degree

Every object in Perv(X) is of finite length, i.e. the category is noetherian and artinian.

◮ It remains to prove that pH0(j!L[dim X]) has finite length. For short exact 0 → L′ → L → L′′ → 0 in Loc(U) we have L′[dim X] → L[dim X] → L′′[dim X]

+1

− − → in D(U) and thus exact sequence pH0(L′[dim X]) → pH0(L[dim X]) → pH0(L′′[dim X]). So the problem is reduced to the case L[dim X] simple. ◮ We have j!L[dim X] is j!∗L[dim X] extended by some object from Perv(Z). The finiteness then follows from induction and that j!∗L[dim X] is simple. ◮ Corollary Let X be a variety of dimension d. For F ∈ pD≥0(X) we have F ∈ D≥−d(X). ◮ For F′ → F → F′′

+1

− − → we have that F′, F′′ ∈ D≥−d(X) = ⇒ F ∈ D≥−d(X). Hence it suffices to prove the assertion for F ∈ Perv(X) simple. But then F ∈ D≥−d(X) is evident from Deligne’s formula.

Perverse sheaves form a stack

◮ Proposition. Suppose we have a Zariski open covering (Ui) of a variety X, Fi ∈ Perv(Ui), isomorphisms αij : Fi|Ui∩Uj

∼

− → Fj|Ui∩Uj such that one Ui ∩ Uj ∩ Uk we have αjk ◦ αij = αik, then they “glue” uniquely to a perverse sheaf F ∈ Perv(X) with natural isomorphisms αi : F|Ui

∼

− → Fi such that αj = αij ◦ αi on Ui ∩ Uj. ◮ Proposition. Similar for morphisms between two perverse sheaves

• n X.

◮ Warning: Neither is true for D(X). ◮ That is, just like U → Sh(U), the functor U → Perv(U) is also a stack of abelian categories on the Zariski site. ◮ Sketch of proof. May assume it’s a cover by two open, as varieties are quasi-compact. Now suppose we have a cover U1, U2 and ja : Ua ֒ → X, j12 : U12 := U1 ∩ U2 ֒ → X. We also have Fa ∈ Perv(Ua) and Then the glueing of sheaves may be constructed as 0 → F → pH0((j1)∗F1) ⊕ pH0((j2)∗F2) → pH0((j12)∗F12)

Proper smooth morphisms and Artin vanishing

◮ Proposition. Suppose f : X → Y is smooth of relative dimension

• d. Then f ∗[d] is t-exact; f ∗G[d] ∈ Perv(X) for any G ∈ Perv(Y ).

◮ Proof. This is really a consequence of our result for f ! for smooth morphisms: f ! = f ∗[2d]. It is obvious for any morphism with fiber dimension ≤ d that f ∗[d] is left t-exact. The only point is then that f ∗[d] is self-dual. ◮ Theorem. (Artin) Let X be an affine variety of dimension d and F a constructible sheaf on X. Then Hk(X; L) = 0 for k > d. ◮ Theorem. (Artin) Let f : X → Y be an affine morphism. Then f∗ is right t-exact, i.e. f∗ sends pD≤0(X) to pD≤0(Y ). ◮ Corollary For an affine quasi-finite morphism j : X → Y , j∗ and j! are t-exact.

Small and semi-small maps

◮ A proper morphism f : X → Y , with say Y equi-dimensional of dimension d, is called small (resp. semi-small) if {y ∈ Y | dim f −1(y) ≥ k} < d − 2k (resp. ≤ d − 2k) for any k ≥ 1. ◮ A semi-small map is necessarily generically finite. Restricting to some Zariski dense open V ⊂ Y we have f : U := f −1(V ) → V finite ´

• etale. Suppose L is a local system on U. Then f∗L is a new local

system on V ; for L ∈ Rep(π1(U)), f∗L = indπ1(V )

π1(U) L ∈ Rep(π1(V )).

◮ Theorem. Suppose L extends to a local system on X, and that X is rationally smooth so that L[d] = IC(X; L). Then f∗L[d] = IC(Y ; f∗(L|U)). ◮ Proof. We claim that i∗f∗L[d] ∈ pD≤−1(Z) where Z = Y − V and i : Z ֒ → Y . Indeed, we have f∗ = f!. In other words, we claim that dimC{z ∈ Z | Hk(f∗L[d])z = 0} < −k for all k ≥ 1. As f∗ = f!, by proper base change this says dimC{z ∈ Z | Hk

c (f −1(z); L[d]|f −1(z)) = 0} < −i, or

dimC{z ∈ Z | Hk

c (f −1(z); L|f −1(z)) = 0} < d − k. But to have Hk c

requires the variety of have dimension at least ⌈ k

2⌉, hence the proof.

Small maps

◮ Example. f : X → Y a small contraction of 3-folds. Then f∗QX[3] = IC(Y ; Q). This gives that H∗(X; Q) = IH(Y ; Q) stated in the beginning of this course. ◮ Example. I will only wave hands in this example1. In Jacquet-Rallis fundamental lemma and its proof by Zhiwei Yun, the conjecture is an identity of two orbital integrals, one for a p-adic GLn and the

• ther for an unramified Un.

◮ After function-sheaf dictionary for ℓ-adic sheaves, the statement is translated to that certain two cohomology are isomorphic. It turns

• ut that by embedding this into a global problem, the two

cohomology in question are the cohomology of some simple sheaves

• n some fibers of two proper morphism from some smooth spaces to

the same space B, which both happen to be small. ◮ So the statement is reduced to that two simple perverse sheaves on B agree, and it suffices to verify this on some generic point, which Yun could compute somewhat directly.

1We refer the interested reader to §4 of Ngˆ

• ’s lecture note Perverse sheaves and

fundamental lemmas.