Semi-small maps Choose V smooth and that F| V has all its cohomology - - PowerPoint PPT Presentation

Semi-small maps

Choose V smooth and that F|V has all its cohomology being local systems, this means to study those supported on Z we look at the local system H− dim V (F|V ).

◮ Now suppose X → Y is semismall, X is rationally smooth of dimension d, and L ∈ Loc(X). We have again f∗L[d] ∈ Perv(Y ) as {y ∈ Y | Hk(f∗L[d])y = 0} ⊂ {y ∈ Y | dim f −1(y) ≥ ⌈ k+d

2 ⌉} has

dimension ≤ d − (k + d) = −k. ◮ To study the constituents of f∗L[d] supported on some Z ֒ → Y irreducible closed, we look at H− dim Z(f∗L[d]|Z). Proper base change gives H− dim Z(f∗L[d]|Z) = Hd−dim Z(f∗(L|f −1(Z))). ◮ That f is semi-small says the generic fiber above Z has dimension at most d−dim Z

2

. Hence Hd−dim Z(f∗(L|f −1(Z))) is the maximum possible degree of the cohomology of the fiber. ◮ Lemma. For an n-dimensional irreducible variety A and L′ ∈ Loc(A) we have H2n

c (A; L′) = L′ π1(A), read as the coinvariant

(L′

x)π1(x,A) for any x ∈ A.

◮ Proof. We may assume A = Asm since the complement only has cohomological dimension ≤ 2n − 2. Poincar´ e-Verdier duality says H2n

c (A; L′) is dual to H0(A; L′∨) = (L′∨)π1(A). Hence the result.

Semi-small maps, II

For an n-dimensional irreducible variety A and L′ ∈ Loc(A) we have H2n

c (A; L′) = L′ π1(A)

◮ By restricting to V ֒ → Z dense open on which f |f −1(V ) → V is Hausdorff locally a product and f∗L is trivial on the base direction, we have Hd−dim Z(f∗(L|f −1(V ))) is the local system whose fiber is

• A(L|A)π1(A) where A runs over irreducible d−dim Z

2

• dimensional

components of the fiber. ◮ This has some kind of monodromy under the base V . Each component of this monodromy in Rep(π(V )) is thus represented by some (simple) constituent of f∗L[d] in Perv(X). ◮ Moreover, suppose we happen to know f∗L[d] is semisimple. Then they are exactly all the constituents of f∗L[d], that is ◮ Theorem Let f : X → Y be a semi-small morphism from X rationally smooth of dimension d, and L ∈ Loc(X). Suppose it is known that f∗L[d] ∈ Perv(Y ) is semisimple. Then f∗L[d] =

• IC(Z; L′)

where Z runs over irreducible closed subsets of dimension d′ and L′ ∈ Rep(π1(V )) is as described above.

BBDG decomposition theorem

◮ Now the big shot, I believe the largest in this course. As a warm-up: ◮ Theorem. (Deligne) Let f : X → Y be proper smooth. Then the spectral sequence Hp(Y ; Rqf∗Q) ⇒ Hn(X; Q) degenerates at E2-page, giving Hn(X; Q) =

• p+q=n

Hp(Y ; Rqf∗Q). ◮ In fact, Deligne proved Rf∗QX = Rqf∗QX[−q] which implies the above. ◮ Definition. The class of simple perverse sheaves of geometric

• rigin is the minimal class of simple perverse sheaves on varieties

that contains

• 1. Constant sheave.
• 2. Simple constituents of pHk(f∗F),pHk(f!F),pHk(f ∗F),pHk(f !F),

pHk(F ⊗ G) and pHk(Hom (F, G)).

◮ Theorem. (Beilinson-Bernstein-Deligne-Gabber) Let f : X → Y be proper and F ∈ Perv(X) be simple of geometric origin. Then f∗F = pHk(f∗F)[−k] in D(Y ) and each pHk(f∗F) ∈ Perv(Y ) is semisimple.

´ etale site

◮ Boring review: a presheaf of abelian groups on a (classical) topological space X is a functor from the category of open sets on X to the category Ab. ◮ With the definition of what counts as an open cover of an open set, sheafification and sheaves make sense. ◮ An ´ etale neighborhood on a scheme (while variety is good enough for us) X is just U → X ´

• etale. An ´

etale cover is a collection of ´ etale neighbborhoods whose images cover the whole X. ◮ The (small) ´ etale site X´

et of a scheme X is the category of ´

etale neighborhoods on X together with the notion of an ´ etale covering. Presheaves and sheaves on them are defined likewise.

Cohomology

◮ Cohomology of a sheaf (resp. hypercohomology of a complex of sheaves) is the derived functor of the global section functor on sheaves on X´

• et. If X is over Spec Fq. Then cohomology can be

identified as Rπ∗ for π : X → Spec Fq. Same for compactly supported cohomology. ◮ Comparison theorem. Let X be a variety over C. Write Xan for the analytic topology on X that we have used for 7 weeks. There is isomorphism H∗(Xan; Z/n) ∼ = H∗(X´

et; Z/n).

◮ However, if we take a torsion free sheaf, e.g. Z, then Hk(X´

et; Z) = 0

for k > 0. For example H1(X´

et; Z) = Hom(π´ et 1 (X), Z) = 0.

◮ So instead, an ℓ-adic sheaf is an inverse system Fn of Z/ℓn sheaves such that Fn ⊗ Z/ℓn−1 = Fn−1 and F = {Fn}. Define H∗(X´

et; F) := lim ← − H∗(X´ et; F) and same for H∗ c . For example

H∗(X´

et; Zℓ) := lim ← − H∗(X´ et; Zℓ). Also put

H∗(X´

et; Qℓ) := H∗(X´ et; Zℓ) ⊗Zℓ Qℓ.

◮ Again, we have H∗(X´

et; Zℓ) ∼

= H∗(Xan; Zℓ).

Constructible sheaves

◮ There is an analogous notion of constructible ℓ-adic complexes. Since all strata in the ´ etale site (whatever that can mean) is constructible, what this means is that we have complexes of Z/{ℓn}-sheaves whose stalks are finite free Z/(ℓn)-modules. ◮ An ℓ-adic sheaf is called torsion if it is killed by ℓn for some n, ◮ The abelian category of Qℓ-sheaves is the quotient category of the category of ℓ-adic sheaves by torsion sheaves. It makes sense to talk about stalks of a Qℓ-sheaf as vector spaces over Qℓ. ◮ Then we have complexes of sheaves which is a mess - one begins with complexes of Z/(ℓn)-sheaves, and ... ◮ Anyhow, in the end, there is a well-behaved notion of bounded derived category D(X´

et) of constructible Qℓ-sheaves on X´ et.

Geometric Frobenius and Weil sheaves

◮ Now “complex” will mean an object in D(X´

et).

◮ Suppose we work with a variety X0 over Fq. Write X := X0 ×Spec Fq Spec Fq. ◮ The relative Frobenius Fr (or sometimes called geometric Frobenius) is the Fq-morphism on X that sends (x1, ...xn) → (xp

1 , ..., xp n ).

◮ For any complex F0 on the (small) ´ etale site of X0, it can be pulled back to a sheaf F on X and that is a canonical isomorphism Fr∗ F ∼ = F. ◮ We have natural maps Fr∗ : H∗(X´

et, F) → H∗(X´ et, Fr∗ F) ∼

= H∗(X´

et, F) and same for

Fr∗ : H∗

c (X´ et, F) → H∗ c (X´ et, Fr∗ F) ∼

= H∗

c (X´ et, F).

◮ More generally, a Weil complex is complex F on X´

et is a complex F

• n X´

et together with an isomorphism F : Fr∗ F ∼

= F. Again we have Fr∗ acting on H∗(X´

et, F) and H∗ c (X´ et, F).

Weights

◮ Now suppose F is a sheaf (not complex) on (X0)´

et, or just X0 for

• short. It is said to be point-wise pure of weight w if Fr∗ acts on

the stalk of F at all geometric points of closed points (i.e. ¯ Fq-points) with eigenvalue of weight k + w. Here a value x ∈ Qℓ is said to be of weight w ∈ Z if it is algebraic over Q and has absolute value qw/2 under any complex embedding. ◮ Same can be said if X0 is a locally of finite type scheme over a finitely generated algebra over Z. ◮ A sheaf F is called mixed of weight ≤ w if there is a filtration for which each graded piece is pure of weight ≤ w. ◮ A complex is mixed of weight ≤ w if each Hk(F) is mixed of weight ≤ k + w. It is called mixed of weight ≥ w if DXF is mixed

• f weight ≤ −w, and called pure of weight w iff it’s both mixed of

weight ≤ w and ≥ w. ◮ Example: Let X0 be a nodal curve. The constant sheaf on X0 is mixed of weight ≤ 0. It is point-wise pure of weight 0, but NOT pure of weight 0.

The Weil conjecture

◮ It’s evident from definition that f ∗ preserves the property of being mixed of weight ≤ w. Dually, f ! preserves the property of being mixed of weight ≥ w ◮ Theorem. Rf! sends those mixed of weight ≤ w to those with the same property. ◮ Corollary. Rf∗ sends those mixed of weight ≥ w to those with the same property. Hence if f is proper, then it preserves the property of being pure of weight w. ◮ Lemma. Let X0 be a variety smooth of dimension d over Fq. Then the constant Weil sheaf QℓX has DXQℓX = QℓX[2d](d), where (d) means twisting the Fr∗ acting by q−d. ◮ Corollary (Weil conjecture) Let X0 be as in the previous lemma and suppose X0 is furthermore proper. Let X = X0 ×Spec Fq Spec Fq as

• before. Then H∗(X; Qℓ) is pure of weight 0.

Pure perverse sheaves

(Weil conjecture) Let X be smooth over ¯ Fq as in the previous lemma and assume X is proper. Then H∗(X´

et; Qℓ) is pure of weight 0.

◮ Let X0 be a variety over Fq, j : U0 ֒ → X0 a Zariski open subset. Let G be a perverse complex on U0. Suppose F is a perverse extension

• f G to X0 such that F has no quotient object on X0 − U0.

◮ Proposition. If G is mixed of weight ≤ w, then so is F. ◮ Corollary. The functor j!∗ preserves weight. In particular ICX0 is always pure of weight −d for a equi-dimensional variety of dimension d. ◮ Corollary. A simple perverse sheaf that is mixed is automatically pure. ◮ Corollary. For X0 proper, we have IH∗(X; Qℓ) is pure of weight 0. ◮ Going back to the decomposition theorem, it really is just the following natural result of Gabber: ◮ Theorem. (Gabber) A pure complex on X = X0 ×Spec Fq Spec Fq is a direct sum of shifted perverse sheaves.