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 | H k ( 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 ) = H d − 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 . Hence H d − dim Z ( f ∗ ( L| f − 1 ( Z ) )) is the maximum 2 possible degree of the cohomology of the fiber. ◮ Lemma. For an n -dimensional irreducible variety A and L ′ ∈ Loc( A ) we have H 2 n c ( A ; L ′ ) = L ′ π 1 ( A ) , read as the coinvariant ( L ′ x ) π 1 ( x , A ) for any x ∈ A . ◮ Proof. We may assume A = A sm since the complement only has cohomological dimension ≤ 2 n − 2. Poincar´ e-Verdier duality says H 2 n c ( A ; L ′ ) is dual to H 0 ( A ; L ′∨ ) = ( L ′∨ ) π 1 ( A ) . Hence the result.

Semi-small maps, II For an n -dimensional irreducible variety A and L ′ ∈ Loc( A ) we have H 2 n c ( A ; L ′ ) = L ′ π 1 ( 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 H d − dim Z ( f ∗ ( L| f − 1 ( V ) )) is the local system whose fiber is A ( L| A ) π 1 ( A ) where A runs over irreducible d − dim Z � -dimensional 2 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 � IC ( Z ; L ′ ) f ∗ L [ d ] = where Z runs over irreducible closed subsets of dimension d ′ and L ′ ∈ Rep( π 1 ( 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 H p ( Y ; R q f ∗ Q ) ⇒ H n ( X ; Q ) degenerates at E 2 -page, giving H n ( X ; Q ) = � H p ( Y ; R q f ∗ Q ) . p + q = n ◮ In fact, Deligne proved Rf ∗ Q X = � R q f ∗ Q X [ − q ] which implies the above. ◮ Definition. The class of simple perverse sheaves of geometric origin is the minimal class of simple perverse sheaves on varieties that contains 1. Constant sheave. 2. Simple constituents of p H k ( f ∗ F ), p H k ( f ! F ), p H k ( f ∗ F ), p H k ( f ! F ), p H k ( F ⊗ G ) and p H k ( H om ( 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 = � p H k ( f ∗ F )[ − k ] in D ( Y ) and each p H k ( 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 F q . Then cohomology can be identified as R π ∗ for π : X → Spec F q . Same for compactly supported cohomology. ◮ Comparison theorem. Let X be a variety over C . Write X an for the analytic topology on X that we have used for 7 weeks. There is isomorphism H ∗ ( X an ; Z / n ) ∼ = H ∗ ( X ´ et ; Z / n ). ◮ However, if we take a torsion free sheaf, e.g. Z , then H k ( X ´ et ; Z ) = 0 for k > 0. For example H 1 ( X ´ et ; Z ) = Hom( π ´ et 1 ( X ) , Z ) = 0. ◮ So instead, an ℓ -adic sheaf is an inverse system F n of Z /ℓ n sheaves such that F n ⊗ Z /ℓ n − 1 = F n − 1 and F = {F n } . Define − H ∗ ( X ´ H ∗ ( X ´ et ; F ) := lim et ; F ) and same for H ∗ c . For example ← H ∗ ( X ´ et ; Z ℓ ) := lim − H ∗ ( X ´ et ; Z ℓ ). Also put ← H ∗ ( X ´ et ; Q ℓ ) := H ∗ ( X ´ et ; Z ℓ ) ⊗ Z ℓ Q ℓ . et ; Z ℓ ) ∼ ◮ Again, we have H ∗ ( X ´ = H ∗ ( X an ; Z ℓ ).

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 / { ℓ n } -sheaves whose stalks are finite free Z / ( ℓ n )-modules. ◮ An ℓ -adic sheaf is called torsion if it is killed by ℓ n for some n , ◮ The abelian category of Q ℓ -sheaves is the quotient category of the category of ℓ -adic sheaves by torsion sheaves. It makes sense to talk about stalks of a Q ℓ -sheaf as vector spaces over Q ℓ . ◮ Then we have complexes of sheaves which is a mess - one begins with complexes of Z / ( ℓ n )-sheaves, and ... ◮ Anyhow, in the end, there is a well-behaved notion of bounded derived category D ( X ´ et ) of constructible Q ℓ -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 X 0 over F q . Write X := X 0 × Spec F q Spec F q . ◮ The relative Frobenius Fr (or sometimes called geometric Frobenius) is the F q -morphism on X that sends ( x 1 , ... x n ) �→ ( x p 1 , ..., x p n ). ◮ For any complex F 0 on the (small) ´ etale site of X 0 , 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 , Fr ∗ F ) ∼ et , F ) → H ∗ ( X ´ = H ∗ ( X ´ et , F ) and same for Fr ∗ : H ∗ et , Fr ∗ F ) ∼ c ( X ´ et , F ) → H ∗ c ( X ´ = H ∗ c ( X ´ et , F ). ◮ More generally, a Weil complex is complex F on X ´ et is a complex F et together with an isomorphism F : Fr ∗ F ∼ on X ´ = 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 ( X 0 ) ´ et , or just X 0 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. ¯ F q -points) with eigenvalue of weight k + w . Here a value x ∈ Q ℓ is said to be of weight w ∈ Z if it is algebraic over Q and has absolute value q w / 2 under any complex embedding. ◮ Same can be said if X 0 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 H k ( F ) is mixed of weight ≤ k + w . It is called mixed of weight ≥ w if D X F is mixed of weight ≤ − w , and called pure of weight w iff it’s both mixed of weight ≤ w and ≥ w . ◮ Example: Let X 0 be a nodal curve. The constant sheaf on X 0 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 X 0 be a variety smooth of dimension d over F q . Then the constant Weil sheaf Q ℓ X has D X Q ℓ X = Q ℓ X [2 d ]( d ), where ( d ) means twisting the Fr ∗ acting by q − d . ◮ Corollary (Weil conjecture) Let X 0 be as in the previous lemma and suppose X 0 is furthermore proper. Let X = X 0 × Spec F q Spec F q as before. Then H ∗ ( X ; Q ℓ ) is pure of weight 0.