The standard filtration on cohomology with compact supports with an appendix on the base change map and the Lefschetz hyperplane theorem
aa r X i v : . [ m a t h . AG ] J a n The standard filtration on cohomologywith compact supportswith an appendix on the base change map andthe Lefschetz hyperplane theorem
Mark Andrea A. de Cataldo
Dedicated to Andrew J. Sommese, on his 60th birthdaywith admiration and respect
Abstract
We describe the standard and Leray filtrations on the cohomology groups withcompact supports of a quasi projective variety with coefficients in a constructiblecomplex using flags of hyperplane sections on a partial compactification of a relatedvariety. One of the key ingredients of the proof is the Lefschetz hyperplane theoremfor perverse sheaves and, in an appendix, we discuss the base change maps for con-structible sheaves on algebraic varieties and their role in a proof, due to Beilinson, ofthe Lefschetz hyperplane theorem.
Contents H c via compactifications . . . . . . . . . . . . . . . . . 11 j ! J ∗ −→ J ∗ j ! . . . . . . . . . . . . . . . . . . . . . 193.3.2 The Lefschetz hyperplane theorem for perverse sheaves . . . . . . . . 203.3.3 A variant of Theorem 3.10 using two hyperplane sections . . . . . . 213.3.4 The Lefschetz hyperplane theorem for constructible sheaves . . . . . 233.4 The generic base change theorem . . . . . . . . . . . . . . . . . . . . . . . . 233.4.1 Statement of the generic base change theorem . . . . . . . . . . . . . 233.4.2 Generic base change theorem and families of hyperplane sections . . 24 Let f : X → Y be a map of algebraic varieties. The Leray filtration on the (hy-per)cohomology groups H ( X, Z ) = H ( Y, Rf ∗ Z X ) is defined to be the standard filtra-tion on H ( Y, Rf ∗ Z X ), i.e. the one given by the images in cohomology of the truncationmaps τ ≤ i Rf ∗ Z → Rf ∗ Z . Similarly, for the cohomology groups with compact supports H c ( X, Z ) = H c ( Y, Rf ! Z X ).D. Arapura’s paper [1] contains a geometric description of the Leray filtration on thecohomology groups H ( X, Z ) for a proper map of quasi projective varieties f : X → Y . Forexample, if Y is affine, then the Leray filtration is given, up to a suitable re-numbering,by the kernels of the restriction maps H ( X, Z ) → H ( X i , Z ) to a suitable collection ofsubvarieties X i ⊆ X . This description implies at once that the Leray filtration, in fact thewhole Leray spectral sequence, is in the category of mixed Hodge structures.The same proof works if we replace the sheaf Z X with any bounded complex C ofsheaves of abelian groups on X with constructible cohomology sheaves. Such complexesare simply called constructible.Since the key constructions take place on Y , given a constructible complex K on Y ,one obtains an analogous geometric description for the standard filtration on H ( Y, K ).For example, if Y is affine, then there is a collection of subvarieties Y i ⊆ Y , obtainedas complete intersections of suitably high degree hypersurfaces in special position, suchthat the standard filtration is given by the kernels of the restriction maps H ( Y, K ) → H ( Y i , K | Y i ).The case of the Leray filtration for a proper map mentioned above is then the specialcase K = Rf ∗ C , and the varieties X i = f − ( Y i ). The properness of the map is usedto ensure, via the proper base change theorem, that the natural base change maps areisomorphisms, so that, in view of the fact that H ( X, C ) = H ( Y, Rf ∗ C ), we can identifythe two maps H ( X, C ) −→ H ( X i , C | X i ) , H ( Y, Rf ∗ C ) −→ H ( Y i , Rf ∗ C | Y i ) , and hence their kernels.We do not know of an analogous description of the Leray filtration on the cohomologygroups H ( X, C ) for non proper maps f : X → Y .2n [1], D. Arapura also gives a geometric description of the Leray filtration on thecohomology groups with compact supports H c ( X, C ) for a proper map f : X → Y of quasiprojective varieties by first “embedding” the given morphism into a morphism f : X → Y of projective varieties, by identifying cohomology groups with compact supports on Y withcohomology groups on Y , and then by applying his aforementioned result for cohomologygroups and proper maps. In his approach, it is important that f is proper, and the identity f ! = f ∗ is used in an essential way.The purpose of this paper is to prove that, given a quasi projective variety Y and aconstructible complex K on Y and, given a (not necessarily proper) map f : X → Y ofalgebraic varieties and a constructible complex C on X , one obtains a geometric descrip-tion of the standard filtration on the cohomology groups with compact supports H c ( Y, K )(Theorem 2.8), and of the Leray filtration on the cohomology groups with compact sup-ports H c ( X, C ) (Theorem 2.9).The proof still relies on the geometric description of the cohomology groups H ( X, C )for proper maps f : X → Y . In fact, we utilize a completion f : X → Y of the varieties and of the map; see diagram (7).For completeness, we include a new proof of the main result of [1], i.e. of the geometricdescription of the Leray filtration on the cohomology groups H ( X, C ) for proper maps f : X → Y ; see Corollary2.3. In fact, we point out that we can extend the result to coverthe case of the standard filtration on the cohomology groups H ( Y, K ); see Theorem 2.2.Theorem 2.2 implies Corollary 2.3.The proof of Theorem 2.2 is based on the techniques introduced in [8], which dealswith perverse filtrations. In the perverse case there is no formal difference in the treatmentof cohomology and of cohomology with compact supports. This contrasts sharply withthe standard case.Even though the methods in this paper and in [8, 6] are quite different from the onesin [1], the idea of describing filtrations geometrically by using hyperplane sections comesfrom [1].In either approach, the Lefschetz hyperplane Theorem 3.14 for constructible sheaveson varieties with arbitrary singularities plays a central role. This result is due to severalauthors, Beilinson [3], Deligne (unpublished) and Goresky and MacPherson [13]. Beilin-son’s proof works in the ´etale context and is a beautiful application of the generic basechange theorem.In the Appendix §
3, I discuss the base change maps for constructible sheaves on al-gebraic varieties and the role played by them in Beilinson’s proof of the Lefschetz hyper-plane theorem. This is merely an attempt to make these techniques more accessible tonon-experts and hopefully justifies the length of this section and the fact that it containsfact well-known to experts.The notation employed in this paper is explained in some detail, especially for non-experts, in § C . We work with bounded complex of sheaves of abelian groups on Y with constructiblecohomology sheaves (constructible complexes, for short) and denote the corresponding3erived-type category by D Y . The results hold, with essentially the same proofs, in thecontext of ´etale cohomology for varieties over algebraically closed fields; we do not discussthis variant. For K ∈ D Y , we have the (hyper)cohomology groups H ( Y, K ) and H c ( Y, K ),the truncated complexes τ ≤ i K and the cohomology sheaves H i ( K ) which fit into the exactsequences (or distinguished triangles)0 −→ τ ≤ i − K −→ τ ≤ i K −→ H i ( K )[ − i ] −→ . Filtrations on abelian groups, complexes, etc., are taken to be decreasing, F i K ⊇ F i +1 K .The quotients (graded pieces) are denoted Gr iF K := F i K/F i +1 K .The standard (or Grothendieck) filtration on K is defined by setting τ p K := τ ≤− p K .The graded complexes satisfy Gr pτ K = H − p ( K )[ p ]. The corresponding decreasing andfinite filtration τ on the cohomology groups H ( Y, K ) and H c ( Y, K ) are called the standard(or Grothendieck) filtrations. Given a map f : X → Y and a complex C ∈ D X , the derivedimage complex Rf ∗ C and the derived image with proper supports complex Rf ! C are in D Y and the standard filtrations on H ( Y, Rf ∗ C ) = H ( X, C ) and H c ( Y, Rf ! C ) = H c ( X, C )are called the Leray filtrations.A word of caution. A key fact used in [8] in the case of the perverse filtration is thatexceptional restriction functors i ! p to general linear sections i p : Y p → Y preserve perversity(up to a shift). This fails in the case of the standard filtration where we must work withhypersurfaces in special position. In particular, i ! of a sheaf is not a sheaf, even aftera suitable shift, and this prohibits the extension of our inductive approach in Theorem2.2 and in Corollary 2.3 from cohomology to cohomology with compact supports. As aconsequence, the statements we prove in cohomology for the standard and and for theLeray filtrations do not have a direct counterpart in cohomology with compact supports,by, say, a reversal of the arrows. The remedy to this offered in this paper passe throughcompletions of varieties and maps.All the results of this paper are stated in terms of filtrations on cohomology groupsand on cohomology groups with compact supports, but hold more generally, and with thesame proofs, for the associated filtered complexes and spectral sequences. However, forsimplicity of exposition, we only state and prove these results for filtrations. Acknowledgments.
I thank Luca Migliorini for many conversations.
In this section we give a geometric description of standard and Leray filtrations on coho-mology and on cohomology with compact supports in terms of flags of subvarieties.
While the paper [8] is concerned with the perverse filtration, its formal set-up is quitegeneral and is easily adapted to the case of the standard filtration. In this section, we4riefly go through the main steps of this adaptation and prove the key Theorem 2.2 andits Corollary 2.3. We refer the reader to [8] for more details and for the proofs of thevarious statements we discuss and/or list without proof.The shifted filtration
Dec ( F ) associated with a filtered complex of abelian groups( L, F ) is the filtration on L defined as follows: Dec ( F ) n L l := { x ∈ F n + l L l | dx ∈ F n + l +1 L l +1 } . The resulting filtrations in cohomology satisfy
Dec ( F ) n H l ( L ) = F n + l H l ( L ) . Proposition 2.1
Let ( L, P, F ) be a bifiltered complex of abelian groups. Assume that H r ( Gr bP Gr aF L ) = 0 ∀ r = a − b. (1) Then L = Dec ( F ) on H ( L ) . Let K ∈ D Y be a constructible complex on a variety Y . By replacing K with asuitable injective resolution, we may assume that K is endowed with a filtration τ suchthat the complex Gr bτ K [ − b ] is an injective resolution of the sheaf H − b ( K ). We take globalsections and obtain the filtered complex ( R Γ( Y, K ) , τ ) for which we have Gr bτ R Γ( Y, K ) = R Γ( Y, H − b ( K )[ b ]). The filtration τ on R Γ( Y, K ) induces the standard filtration (denotedagain by τ ) on the cohomology groups H ∗ ( Y, K ).An n -flag on Y is an increasing sequence of closed subspaces Y ∗ : ∅ = Y − ⊆ Y ⊆ . . . ⊆ Y n = Y. The flag Y ∗ induces a filtration F Y ∗ on K as follows: (recall that j ! = Rj ! and k ! = Rk ! are extension by zero) set j a : Y \ Y a − → Y and define F aY ∗ K := j a ! j ∗ a K = K Y − Y a . Setting k a : Y a \ Y a − → Y , we have Gr aF K = k a ! k ∗ a K = K Y a − Y a − . The correspondingfiltration in cohomology is F aY ∗ H r ( Y, K ) = Ker { H r ( Y, K ) −→ H r ( Y a − , K | Y a − ) } . Taking global sections, we get the filtered complex ( R Γ( Y, K ) , F Y ∗ ) with the propertythat Gr aF Y ∗ R Γ( Y, K ) = R Γ( Y, K Y a − Y a − ) = R Γ( Y a , ( K | Y a ) Y a − Y a − ) . (2)We have Gr bτ Gr aF Y ∗ R Γ( Y, K ) = R Γ( Y a , H − b ( K )[ b ] Y a − Y a − ) , so that H r ( Gr bτ Gr aF Y ∗ R Γ( Y, K )) = H r + b ( Y a , ( H − b ( K ) | Y a ) Y a − Y a − ) . (3)5ote that the left-hand-side is the relative cohomology group H r + b ( Y a , Y a − , H − b ( K ) | Y a ) = H r + b ( Y a , j a ! j ∗ a H − b ( K ) | Y a ) , (4)where j a : Y a \ Y a − → Y a . This is important in what follows as it points to the use wenow make of the Lefschetz hyperplane theorem for Sheaves 3.14. Theorem 2.2
Let Y be an affine variety of dimension n and K ∈ D Y be a constructiblecomplex on Y . There is an n -flag Y ∗ on Y such that τ = Dec ( F Y ∗ ) on H ( Y, K ) . Proof.
The goal is to choose the flag Y ∗ so that (1) holds when ( L, P, F ) := ( R Γ( Y, K ) , τ, F Y ∗ ).In view of (4), we need the flag to satisfy the condition H r ( Y a , j a ! j ∗ a ( H β ( K )) | Y a ) = 0 ∀ r = a, ∀ a ∈ [0 , n ] , ∀ β. (5)Note that Theorem 3.14 applies to any finite collection of sheaves (in fact it applies to anycollection of sheaves which are constructible with respect to a fixed stratification). Theflag is constructed by descending induction on the dimension of Y . By definition, Y n = Y .It is sufficient to choose Y n − as in Theorem 3.14. We repeat this process, replacing Y n with Y n − and construct the wanted flag inductively.Let f : X → Y be a map of algebraic varieties with Y affine and C ∈ D X . TheLeray filtration L f on H ( X, C ) = H ( Y, Rf ∗ C ) is, by definition, the standard filtration τ on H ( Y, Rf ∗ C ). Theorem 2.2 yields an n -flag Y ∗ on Y such that L f = Dec ( F Y ∗ ). In theapplications though, it is more useful to have a description in terms of a flag on X . Let X ∗ := f − Y ∗ be the pull-back flag on X , i.e. X a := f − Y a . There is the commutativediagram H ( X, π ∗ C ) r (cid:15) (cid:15) = H ( Y, Rf ∗ π ∗ C ) r ′ (cid:15) (cid:15) H ( X a , i ∗ a π ∗ C ) = H ( Y a , Rf ∗ i ∗ a π ∗ C ) H ( Y a , i ∗ a Rf ∗ π ∗ C ) , b o o (6)where b stems from the base change map (20) i ∗ a Rf ∗ C → Rf ∗ i ∗ a C. The kernels of thevertical restriction maps r and r ′ define the filtrations F X ∗ and F Y ∗ . It is clear thatKer r ⊇ Ker r ′ , i.e. that F X ∗ ⊇ F Y ∗ , and that equality holds if the base change map is anisomorphism.The following is now immediate. Corollary 2.3
Let f : X → Y be a proper map with Y affine of dimension n and C ∈ D X .There is an n -flag X ∗ on X such that L f = Dec ( F X ∗ ) on H ( X, C ) . roof. Since f is proper, the base change map i ∗ a Rf ∗ C → Rf ∗ i ∗ a C is an isomorphism.In this section we have proved results for when Y is affine. In this case the statementsand proofs are more transparent and the flags are on Y (pulled-back from Y in the Leraycase). The case when Y quasi projective case is easily reduced to the affine case in thenext section. In this section we extend the results of the previous section from the case when Y isaffine, to the case when Y is quasi projective. The only difference is that, given a quasiprojective variety Y , we need to work with an auxiliary affine variety Y which is a fiberbundle, π : Y → Y , over Y with fibers affine spaces A d and we need the flag to be an( n + d )-flag Y ∗ on Y . This construction is due to Jouanolou.Here is one way to prove this. In the case Y = P n , take Y := ( P n × P n ) \ ∆ with π either projection. In general, take a projective completion Y ′ of Y . Blow up the boundary Y ′ \ Y and obtain a projective completion Y of Y such that Y → Y is affine. Embed Y in some P N . Take the restriction of the bundle projection ( P N × P N ) \ ∆ → P N over Y toobtain the desired result.Let Y be a quasi projective variety. We fix a “Jouanolou fibration” π : Y → Y as above.If Y is affine, then we choose the identity. In order to distinguish standard filtrations ondifferent spaces, e.g. Y and Y , we occasionally write τ Y and τ Y . Since the fibers of π arecontractible, we have canonical identifications of filtered groups( H ( Y, K ) , τ Y ) = ( H ( Y , π ∗ K ) , τ Y ) . This identity, coupled with Theorem 2.2 yields at once the following
Theorem 2.4
Let Y be quasi projective of dimension n and K ∈ D Y . There is an ( n + d ) -flag Y ∗ on Y such that ( H ∗ ( Y, K ) , τ Y ) = ( H ∗ ( Y , K ) , Dec ( F Y ∗ )) , i.e. τ pY H r ( Y, K ) = Ker { π ∗ p + r − : H r ( Y, K ) −→ H r ( Y p + r − , π ∗ p + r − K ) } . If Y is affine, then Y = Y and the flag is an n -flag on Y . In order to generalize Corollary 2.3 about the Leray filtration, we form the Cartesiandiagram (where maps of the “same” type are denoted with the same symbol) X f / / π (cid:15) (cid:15) Y π (cid:15) (cid:15) X f / / Y H ( X , π ∗ C ) = H ( X, C ) = H ( Y, Rf ∗ C ) = H ( Y , π ∗ Rf ∗ C ) = H ( Y , Rf ∗ π ∗ C ))and we have the identity of the corresponding filtrations L f : X →Y = L f : X → Y = τ Y = τ Y = τ Y . Theorem 2.5
Let f : X → Y be a proper map of algebraic varieties, let Y be quasiprojective and C ∈ D X . There is an ( n + d ) -flag X ∗ on X such that ( H ( X, C ) , L f ) = ( H ( X , π ∗ C ) , Dec ( F X ∗ )) , i.e. L p H r ( X, C ) = Ker { π ∗ p + r − : H r ( X, C ) −→ H r ( X p + r − , π ∗ p + r − C ) } . Proof.
We have L f : X → Y = L f : X →Y so that we may replace f : X → Y with f : X → Y .Since Y is affine, we can apply Corollary 2.3 and conclude. Remark 2.6
If the map f is not proper, then the relevant base change map is not anisomorphism. While it is possible to describe the Leray filtration using a flag on Y (on Y if Y is affine), we do not know how to describe it using a flag on X (on X if Y is affine).This latter description would be more desirable in view, for example, of the followingHodge-theoretic application due to Arapura [1]. We also do not know how to do so usingcompactifications; see Remark 2.12. Corollary 2.7
Let f : X → Y be a proper morphism with Y quasi projective. Then theLeray filtration on H ( X, Z ) is by mixed Hodge substructures.Proof. The Leray filtration satisfies L f = Dec ( F Y ∗ ) for some flag on the auxiliary space Y . Since the base change maps are isomorphisms, F Y ∗ = F X ∗ . By the usual functorialityproperty of the canonical mixed Hodge structures on varieties, the latter filtration is givenby mixed Hodge substructures of H ( X , Z ) = H ( X, Z ). Since sheaves do not behave well with respect to Verdier Duality, it is not possible todualize the results in cohomology to obtain analogous ones for cohomology with compactsupports.Given a map of varieties f : X → Y and C ∈ D X , the Leray filtration L f : X → Y on H c ( X, C ) = H c ( Y, Rf ! C ) is defined to be the standard filtration on the last group.In this section we give a geometric description of the standard and of the Leray filtra-tions on cohomology with compact supports. The description of the Leray filtration onthe cohomology groups with compact supports H c ( X, C ) is valid for any (not necessarily8roper) map, and this is in contrast with the case of the cohomology groups H ( X, C ) (seeRemark 2.6).Arapura’s [1] proves an analogous result for proper maps, but to our knowledge, thatmethod does not extend to non proper maps. Nevertheless, the method presented here isclose in spirit to Arapura’s.The method consists of passing to completion of varieties and maps and then use thebase change properties associated with these compactifications to reduce to the case ofcohomology and proper maps. One main difference with cohomology is that, even if westart with Y affine, the flag is on the auxiliary space Y (see below). We use freely the fact, due to Nagata, that varieties and maps can be compactified, i.e. anyvariety Y admits an open immersion into a complete variety with Zariski dense image, andgiven any map f : X → Y , there are a proper map f ′ : X ′ → Y and an open immersion X → X ′ with Zariski-dense image such that f ′| X = f .Since our result are valid without the quasi projectivity assumption on X , we invokeNagata’s deep result. If X and Y are both quasi projective, then it is easy to get by takingprojective completions of X and Y and by resolving the indeterminacies.Let f : X → Y be a map with Y quasi projective. Choose a projective completion j : Y → Y such that j is an affine open embedding. This can be achieved by first takingany projective completion and then by blowing up the boundary. Choose an A d -fibration π : Y → Y . Choose a completion j : X → X such that f extends to a (necessarily) proper f : X → Y .
Choose closed embeddings i : Y a → Y , e.g. the constituents of a flag Y ∗ on Y . There is the following commutative diagram, where the squares and parallelogramslabelled (cid:13) , . . . , (cid:13) a j / / i (cid:15) (cid:15) f (cid:22) (cid:22) ----------------------------------------------- X ai (cid:15) (cid:15) f (cid:8) (cid:8) (cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17) (cid:13) X j / / π (cid:15) (cid:15) (cid:13) f (cid:22) (cid:22) ------------------------------------------------ X π (cid:15) (cid:15) (cid:13) f (cid:8) (cid:8) (cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17) (cid:13) X j / / (cid:13) f (cid:22) (cid:22) ------------------------------------------------ X (cid:13) f (cid:8) (cid:8) (cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17) Y a j / / i (cid:15) (cid:15) Y ai (cid:15) (cid:15) (cid:13) Y j / / π (cid:15) (cid:15) Y π (cid:15) (cid:15) (cid:13) Y j / / Y . (7)If f : X → Y is not proper, then the commutative trapezoids are not Cartesian.Note that since j : Y → Y and Y are affine, the map π : Y → Y is an A d -bundle withaffine total space Y . We shall use freely the facts that follow.1. Due to the smoothness of the maps π and the properness of the maps f , the basechange Theorem holds for (cid:13) , (cid:13) , (cid:13) , (cid:13) (cid:13) .
2. For the remaining Cartesian squares (cid:13) , (cid:13) , and (cid:13) i ∗ Rj ∗ −→ Rj ∗ i ∗ , i ∗ Rf ∗ −→ Rf ∗ i ∗ . (8)10. The exactness of j ! implies that R j ! = j ! is simply extension by zero and that itcommutes with ordinary truncation, i.e. j ! ◦ τ ≤ i = τ ≤ i ◦ j ! ; similarly, for the formationof cohomology sheaves. The compactness of Y and X implies that, H ( Y , − ) = H c ( Y , − ), etc. Recall that H c ( Y, − ) = H c ( Y , j ! ( − )). It follows that we have, forevery K ∈ D Y , canonical identifications of filtered groups( H c ( Y, K ) , τ Y ) = ( H c ( Y , j ! K ) , τ Y ) = ( H ( Y , j ! K ) , τ Y ) . (9)If K ∈ D Y is any extension of K ∈ D Y to Y , e.g. j ! K , j ∗ K etc., then we also have( H c ( Y, K ) , τ Y ) = ( H ( Y , j ! j ! K ) , τ Y ) . (10)Similarly, for the other open immersions j in diagram (7). In view of the definition ofrelative cohomology as the hypercohomology of j ! j ! ( − ), we also have H ( Y , j ! j ! K ) =( H ( Y , Y \ Y ; K ).4. There are canonical identifications:( H ( Y, K ) , τ Y ) = ( H ( Y , π ∗ K ) , τ Y ) . (11)Similarly, for the other maps π.
5. There are canonical identifications:( H ( X, C ) , L fτ ) = ( H ( Y, f ∗ C ) , τ Y ) = ( H ( Y , π ∗ f ∗ C = f ∗ π ∗ C ) , τ Y ) = ( H ( X , π ∗ C ) , L fτ ) . (12)Similarly, for the f in (cid:13) F , the filtration F ( l ) is defined by setting F ( l ) i := F l + i ).Since π ∗ = π ! [ − d ] is an exact functor, we have canonical identifications of filteredgroups( H c ( X, C ) , L fτ ) = ( H c ( Y, f ! C ) , τ Y ) = ( H c ( Y , π ! f ! C = f ! π ! C ) , τ Y (2 d )) . (13)Similarly, for the other maps of type f and f . H c via compactifications Let Y be a quasi projective variety of dimension n and K ∈ D Y be a constructible complexon Y . Consider any diagram as in (7). (cid:13)
1. We can choose Y to be of dimension n . Theorem 2.8
There is an ( n + d ) -flag Y ∗ on Y for which we have the following identityof filtered groups ( H c ( Y, K ) , τ Y ) = H ( Y , π ∗ j ! K ) , Dec ( F Y ∗ )) . roof. In view of (9) and of (11) applied to π : Y → Y , we have canonical identificationsof filtered groups( H c ( Y, K ) , τ Y ) = ( H ( Y , j ! K ) , τ Y ) = ( H ( Y , π ∗ j ! K ) = H ( Y , j ! π ∗ K ) , τ Y ) . (14)The conclusion follows from Theorem 2.4 applied to the pair ( Y , j ! K ).Let f : X → Y be a map of algebraic varieties and C ∈ D X . Consider any diagram asin (7). We can choose Y to be of dimension n . There are natural identifications of groups H c ( X, C ) = H c ( X, j ! C ) = H ( X, j ! C ) = H ( X , π ∗ j ! C = j ! π ∗ C ) . (15)We endow these groups with the respective Leray filtrations. Note that since f is proper,the Leray filtration on H c ( X, j ! C ) coincides with the ones for H ( X, j ! C ). Theorem 2.9
There is an ( n + d ) -flag X ∗ on X for which we have the following identityof filtered groups L f : X → Yτ = L f : X → Yτ = Dec ( F Y ∗ ) = Dec ( F X ∗ ) , on H c ( X, C ) . (16) Proof.
The filtration L f : X → Y is the standard filtration on H c ( Y, Rf ! C ) which in turn, bythe exactness of j ! and the equality H c ( Y, − ) = H c ( Y , j ! ( − )), coincides with the standardfiltration on H c ( Y , j ! Rf ! C ). Since Y is compact and f is proper (so that Rf ! = Rf ∗ ), bythe commutativity of the base trapezoid diagram in (7), we have that Rf ! j ! = j ! Rf ! sothat H c ( Y , j ! Rf ! C ) = H ( Y , Rf ∗ j ! C ). This implies the equality L f : X → Yτ = L f : X → Yτ . Weare now in the realm of cohomology and proper maps and the rest follows from Theorem2.5 applied to f and to j ! C . Corollary 2.10
The Leray filtration on H c ( X, Z ) is by mixed Hodge substructures.Proof. By Theorem 2.9 and (10), the filtration in question is the one induced by the flag X ∗ on the relative cohomology group H ( X , j ! Z ) = H ( X , X , Z ). The result follows fromDeligne’s mixed Hodge Theory [11]. Remark 2.11
The case when f is proper is proved in [1]. Remark 2.12
If one tries to imitate the procedure we have followed in the case of thecohomology groups with compact supports H c ( X, C ) for an arbitrary map f : X → Y ,with the goal of obtaining an analogous result for the Leray filtration on the cohomologygroups H ( X, C ), then one hits the following obstacle: indeed, there are identifications H ( Y, Rf ∗ C ) = H ( Y , Rj ∗ Rf ∗ C ) = H ( Y , Rf ∗ Rj ∗ C ), however, since Rj ∗ does not commutewith truncation, the Leray filtrations for f and f do not coincide, and the imitation of theprocedure would yield a geometric description only for the case of f .12 Appendix: Base change and Lefschetz hyperplane theo-rem
Varieties and maps. A variety is a separated scheme of finite type over the fieldof complex numbers C . In particular, we do not assume that varieties are irreducible,reduced, or even pure dimensional. Since we work inductively with intersections of specialhypersurfaces, we need this generality even if we start with a nonsingular irreduciblevariety. A map is a map of varieties, i.e. map of C -schemes. Coefficients.
The results of this paper hold for sheaves of R -modules, where R is acommutative ring with identity with finite global dimension, e.g. R = Z , R a field, etc.For the sake of exposition we work with R = Z , i.e. with sheaves of abelian groups. Variants.
The results of this paper hold, with routine adaptations of the proofs, inthe case of varieties over an algebraically closed field and ´etale sheaves with the usualcoefficients: Z /l m Z , Z l , Q l , Z l [ E ], Q l [ E ] ( E ⊇ Q l a finite extension) and Q l . These variantsare not discussed further (see [4], § § Stratifications.
The term stratification refers to an algebraic Whitney stratification[5, 12, 13]. Recall that any two stratifications admit a common refinement and that mapsof varieties can be stratified. See also § The constructible derived category D Y . Let Y be a variety, Sh Y be the abeliancategory of sheaves of abelian groups on Y and D ( Sh Y ) be the associated derived category.A sheaf F ∈ Sh Y is constructible if there is a stratification of Y such that the restrictionof F to each stratum is locally constant with stalk a finitely generated abelian group. Acomplex is bounded if the cohomology sheaves H i ( K ) = 0 for | i | ≫
0. A complex K ∈ D ( Sh Y ) with constructible cohomology sheaves is said to be constructible . The category D Y = D Y ( Z ) is the full subcategory of the derived category D ( Sh Y ) whose objects arethe bounded constructible complexes. For a given stratification Σ of Y , a complex withthis property is called Σ-constructible. Given a stratification Σ of Y , there is the fullsubcategory D Σ Y ⊆ D Y of complexes which are Σ-constructible. Hypercohomology groupsare denoted H ( Y, K ) and H c ( Y, K ). If K ∈ D Y and n ∈ Z , then K [ n ] ∈ D Y is the ( n -shifted ) complex with ( K [ n ]) i = K i + n . One has, for example, H i ( Y, K [ n ]) = H i + n ( Y, K ). The four functors associated with a map . Given a map f : X → Y , there are theusual four functors ( f ∗ , Rf ∗ , Rf ! , f ! ). By abuse of notation, denote Rf ∗ and Rf ! simplyby f ∗ and f ! . The four functors preserve stratifications, i.e. if f : ( X, Σ ′ ) → ( Y, Σ) isstratified, then f ∗ , f ! : D Σ ′ X → D Σ Y and f ∗ , f ! : D Σ Y → D Σ ′ X . Verdier Duality.
The Verdier Duality functor D = D Y : D Y → D Y is an autoequiv-alence with D ◦ D = Id D Y and it preserves stratifications. We have D Y f ! = f ∗ D X and D X f ! = f ∗ D Y . Perverse sheaves.
We consider only the middle perversity t -structure on D Y [4].There is the full subcategory P Y ⊆ D Y of perverse sheaves on Y . The elements are special13omplexes in D Y . An important example is the intersection complex of an irreduciblevariety [12, 5]. Let j : U → Y be an open immersion; then j ! = j ∗ : P Y → P U ,i.e. they preserve perverse sheaves. Let j : U → Y be an affine open immersion; then j ! , j ∗ : P U → P Y . The Verdier Duality functor D : P Y → P Y is an autoequivalence. Distinguished triangles for a locally closed embedding.
There is the notionof distinguished triangle in D Y : it is a sequence of maps X → Y → Z → X [1] which isisomorphic in D Y to the analogous sequence of maps arising from the cone constructionassociated with a map of complexes X ′ → Y ′ . Let j : U → Y be a locally closed embeddingwith associated “complementary” embedding and i : Y \ U → Y . For every K ∈ D Y , wehave distinguished triangles j ! j ! K −→ K −→ i ∗ i ∗ K [1] −→ , i ! i ! K −→ K −→ j ∗ j ∗ K [1] −→ . (17) Various base change maps.
Given two maps Y ′ g → Y f ← X, there is the Cartesiandiagram X ′ g / / f (cid:15) (cid:15) X f (cid:15) (cid:15) Y ′ g / / Y. (18)The ambiguity of the notation (clearly the two maps g are different from each other, etc.)does not generate ambiguous statements in what follows, and it simplifies the notation.There are the natural maps g ! f ∗ −→ f ∗ g ! , g ∗ f ! −→ f ! g ∗ . (19)There are the base change maps g ∗ f ∗ −→ f ∗ g ∗ , g ! f ! ←− f ! g ! (20)and the base change isomorphisms g ∗ f ! ≃ −→ f ! g ∗ , g ! f ∗ ≃ ←− f ∗ g ! . (21)Similarly, for the higher direct images R i f ∗ and R i f ! . Example 3.1
Let Y ′ → Y be the closed embedding of a point y → Y . The first basechange map in (20) yields a map ( R i f ∗ Z X ) y → H i ( f − ( y ) , Z ). Given a sufficiently small,contractible neighborhood of y in Y , we have H i ( f − ( U y ) , Z ) = ( R i f ∗ Z X ) y . This basechange map is seldom an isomorphism, e.g. the open immersion X = C ∗ → C = Y , y = 0.This failure is corrected in (21) by taking the direct image with proper supports. Base change theorems.
The base change maps (20) are isomorphisms if either oneof the following conditions is met: f is proper, f is locally topologically trivial over Y , or g is smooth. 14 he octahedron axiom. This is one of the axioms for a triangulated category andcan be found in [4], 1.1.6. Here is a convenient way to display it (see [4], 1.1.7.1). Givena composition X f → Y g → Z of morphisms one has the following diagram Z ′ AAAAAAAA Y > > ~~~~~~~~ g / / Z / / ( ( PPPPPPPPPPPPPPP Y ′ BBBBBBBB X f ? ? ~~~~~~~~ gf ooooooooooooooo X ′ (22)where ( X, Y, Z ′ ) , ( Y, Z, X ′ ) , ( X, Z, Y ′ ) and ( Z ′ , Y ′ , X ′ ) are distinguished triangles. Remark 3.2
It is clear that g is an isomorphism if and only if Y ′ ≃ X ′ → Z ′ [1] is an isomorphism. The term “general.”
Let P be a property expressed in terms of the hyperplanes ofa projective space P , i.e. the elements of P ∨ . We say that property P holds for a generalhyperplane if there is a Zariski-dense open subset V ⊆ P ∨ such that property P holds forevery hyperplane in V . Of course this terminology applies to propositions “parameterized”by irreducible varieties and one can talk about a general pair of hyperplanes, in which casethe variety is P ∨ × P ∨ , etc.As it is customary, we often denote a canonical isomorphism with the symbol “=.” Let us discuss the following two special cases of (18). Even though (23) is a special caseof (24), it is convenient to distinguish between the two (see Propositions 3.4, 3.5). Let i : H → Y be a closed embedding. Let j : U → Y be an open embedding and f : X → Y be a map. We obtain the following two Cartesian diagrams U ∩ H i / / j (cid:15) (cid:15) U j (cid:15) (cid:15) H i / / Y, (23) X H i / / f (cid:15) (cid:15) X f (cid:15) (cid:15) H i / / Y. (24)15 uestion 3.3 Let K ∈ D Y , C ∈ D X . Which conditions on the closed embedding i : H → Y ensure that the base change maps j ∗ i ∗ K −→ i ∗ j ∗ K, j ! i ! K ←− i ! j ! K are isomorphisms?Which conditions on the closed embedding i : X H → X ensure that the base change maps f ∗ i ∗ K −→ i ∗ f ∗ K, f ! i ! K ←− i ! f ! K are isomorphisms?Answers to these questions are given in Propositions 3.4, 3.5. Since these results involvethe notion of stratifications, we discuss briefly stratifications in the next section. For background, see [5, 13]. The datum of a stratification Σ of thevariety Z includes a disjoint union decomposition Z = ` Σ i into locally closed nonsingular irreducible subvarieties Σ i called strata . One requires that the closure of a stratum is aunion of strata. These data are subject to the Whitney Conditions A and B, whichwe do not discuss here. Every variety admits a stratification. Any two stratificationsof the same variety admit a common refinement. Given a stratification Σ of Z , everypoint z ∈ Z admits a fundamental system of standard neighborhoods homeomorphic, in astratum-preserving-way, to C l × C ( L ), where C denotes the real cone (with vertex v ), L isthe link of z in Z (relative to Σ) (it is a stratified space obtained by embedding Z in somemanifold, intersecting Z with a submanifold meeting the stratum transversally at z andthen intersecting the result with a small ball centered at z ), and C l × v is the intersectionof the stratum Σ i to which z belongs with a small ball (in the big manifold containing Z )centered at z . Constructible complexes.
Let Σ be a stratification of the variety Z . The boundedcomplexes constructible with respect to Σ form the category D Σ Z which is a full subcategoryof the constructible derived category D Z . If K ∈ D Σ Z , z ∈ Z , U := C l × C ( L ) is a standardneighborhood of z with second projection π , then K | U ≃ π ∗ π ∗ K | U , i.e. K is locally apull-back from the cone over the link. Stratified maps.
Algebraic maps can be stratified: given a map f : X → Y , thereare stratifications Σ X for X and Σ Y for Y such that (i) for every stratum S on Y , thespace f − S is a union of strata on X and (ii) for y ∈ S there exists a neighborhood U of y in S , a stratified space F and a stratification-preserving homeomorphism F × U ≃ f − U which transforms the projection onto U into f . If f is a closed embedding, then eachstratum in X is the intersection of X with a stratum of Y of the same dimension. If f is an open immersion, using standard neighborhoods, the local model at y ∈ Y \ X is f : C l × ( C ( L ) − C ( L ′ )) → C l × C ( L ), where L ′ is the link at z of Y \ X .16 ormally nonsingular inclusions. A closed embedding i : H → Y is normallynonsingular with respect to a stratification Σ of Y if H is obtained locally on Y by thefollowing procedure: embed Y into a manifold M and H is the intersection H ′ ∩ Y , where H ′ ⊆ M is a submanifold meeting transversally all the strata of Σ. See [12, 7]. Notethat a normally nonsingular inclusion is locally of pure codimension. If Y is embeddedinto some projective space, then by the Bertini Theorem a general hyperplane sectionyields a normally nonsingular inclusion. More generally, the general element of a finitedimensional base-point-free linear system of Y yields a normally nonsingular inclusion([14]). Let K ∈ D Σ Y and i : H → Y be a normally nonsingular inclusion of complexcodimension r with respect to Σ. Then i ∗ K = i ! K [ − r ] (cf. [7]). Let j : U → Y be an open embedding and Σ be a stratification of Y such that, if Σ U is itstrace on U , then the map j : ( U, Σ U ) → ( Y, Σ) is stratified. Such a stratification Σ exists.Consider the situation (23).
Proposition 3.4
Assume that i : H → Y is normally nonsingular with respect to Σ .Then for every K ∈ D Σ U U the base change maps i ∗ j ∗ K −→ j ∗ i ∗ K, i ! j ! K ←− j ! i ! K are isomorphisms.Proof. Here are two essentially equivalent proofs. While the first one seems shorter, itdoes rely on the formula i ∗ = i ! [ − r ], the second one is more direct.1 st proof. The assumptions imply that i ∗ K = i ! K [ − r ]. The conclusion follows from thebase change isomorphisms (21).2 nd proof. The complexes j ! K, j ∗ K ∈ D Σ Y . The assertion is local. The local model for (23)at a point y ∈ H lying on a l -dimensional stratum with links L for y ∈ Y and L ′ ⊆ L for y ∈ Y \ U is, denoting by C real cones and by r the codimension of H in Y : C l − r × ( C ( L ) \ C ( L ′ )) i / / j (cid:15) (cid:15) C l × ( C ( L ) \ C ( L ′ )) π / / j (cid:15) (cid:15) { y } × ( C ( L ) \ C ( L ′ )) j (cid:15) (cid:15) C l − r × C ( L ) i / / C l × C ( L ) π / / { y } × C ( L ) (25)with Id ≃ π ∗ π ∗ = π ! π ! for Σ-constructible complexes. One has, using the base changeTheorem for the smooth map π ◦ i : i ∗ j ∗ K ≃ i ∗ π ∗ π ∗ j ∗ K = i ∗ π ∗ j ∗ π ∗ K = j ∗ i ∗ π ∗ π ∗ K ≃ j ∗ i ∗ K. This proves the first assertion. The second one is proved in the same way: i ! j ! K ≃ i ! π ! π ! j ! K = i ! π ! j ! π ! K = j ! i ! π ! π ! K ≃ j ! i ! K. K ∈ D Σ U U , we have that K ∨ ∈ D Σ U U . We have proved that the first assertion holdsfor every K ∈ D Σ U U so that it holds for K ∨ : i ∗ j ∗ K ∨ ≃ j ∗ i ∗ K ∨ and the second assertionfollows by applying Verdier Duality to this isomorphism.Consider the Cartesian diagram (24) and let Σ ′ be a stratification of X . Proposition 3.5
Let i : X H → X be a normally nonsingular inclusion with respect to Σ ′ .Then for every C ∈ D Σ ′ X the base change maps i ∗ f ∗ C −→ f ∗ i ∗ C, i ! f ! C ←− f ! i ! C are isomorphisms.Proof. Let j : X → X f → Y be a completion of the map f, i.e. j is an open immersionwith dense image and f is proper. Such a completion exists by a fundamental result ofNagata. The Cartesian diagram (24) can be completed to a commutative diagram withCartesian squares X H i / / j " " DDDDDDDD f (cid:15) (cid:15) X j (cid:31) (cid:31) ???????? f (cid:15) (cid:15) X H i / / f (cid:4) (cid:4) (cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) X f (cid:6) (cid:6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H i / / Y . (26)By virtue of Lemma 3.4, we have the base change isomorphism i ∗ j ∗ C ≃ j ∗ i ∗ C to whichwe apply the proper map f : X H → H : f ∗ i ∗ j ∗ C ≃ f ∗ j ∗ i ∗ C = f ∗ i ∗ C. The first assertion follows by applying the Base Change for Proper Maps to the first term: f ∗ i ∗ j ∗ C = i ∗ f ∗ j ∗ C = i ∗ f ∗ C. The second one can be proved in either of two ways as in the proof of Proposition 3.4.
Remark 3.6
Note that X H can be normally included in X with respect Σ ′ while H mayfail to be so with respect to any stratification of Y for f ∗ C and f ! C . E.g. X is nonsingular, C = Z X , but Y and/or f have singularities.18 .3 The Lefschetz hyperplane theorem The classical Lefschetz hyperplane theorem states that if Y is a projective manifold ofdimension n and H is a smooth hyperplane section relative to an embedding into projectivespace, then the restriction map H i ( Y, Z ) → H i ( H, Z ) is an isomorphism for i < n − i = n − H i ( Y, H, Z ) = 0 for i ≤ n − j : Y \ H → Y is the open embedding, then j ∗ = j ! and, since Y is compact, H i ( Y, H, Z ) = H ( Y, j ! j ∗ Z Y ). We can thus reformulate the Lefschetz hyperplane theoremin terms of the following vanishing statement H i ( Y, j ! j ! Z [ n ]) = 0 ∀ i < . Note that Z [ n ] is a perverse sheaf on the nonsingular Y . Beilinson [3], Lemma 3.3, hasgiven a proof of this important result which is valid in the ´etale case and for every perversesheaf on a quasi projective variety Y . His proof is based on the natural map (32) beingan isomorphism.In this section, we discuss Beilinson’s proof, which boils down to an application of thebase change Proposition 3.4. j ! J ∗ −→ J ∗ j ! Let Y be a quasi projective variety, Y ⊆ P N be a fixed embedding in some projectivespace, Y ⊆ P N be the closure of Y , Λ ⊆ P N be a hyperplane and H ⊆ Y and H ⊆ Y bethe corresponding hyperplane sections. There is the Cartesian diagram H i / / J (cid:15) (cid:15) Y J (cid:15) (cid:15) U J (cid:15) (cid:15) j o o H i / / Y U . j o o (27)Let K ∈ D Y . Consider the composition J ∗ K −→ i ∗ i ∗ J ∗ K φ −→ i ∗ J ∗ i ∗ K (= J ∗ i ∗ i ∗ K ) . (28)The octahedron axiom yields a distinguished triangle (the equality stems from (21)) j ! j ! J ∗ K [1](= j ! J ∗ j ! K [1]) −→ J ∗ j ! j ! K [1] −→ Cone( φ ) −→ , (29)where the first map arises by applying (19) to j ! K .Similarly, we have the composition J ! K ←− i ! i ! J ! K ϕ ←− i ! J ! i ! K (= J ! i ! i ! K ) (30)19nd the octahedron axiom, yields a distinguished triangle ←− j ∗ j ∗ J ! K (= j ∗ J ! j ∗ K ) ←− J ! j ∗ j ∗ K ←− Cone( ϕ ) , (31)where the second map arises by applying (19) to j ∗ K . Lemma 3.7
The map j ! J ∗ j ! K −→ J ∗ j ! j ! K ( j ∗ J ! j ∗ K ←− J ! j ∗ j ∗ K, resp. ) (32) is an isomorphism if and only if the base change map i ∗ J ∗ K −→ J ∗ i ∗ K ( i ! J ! K ←− J ! i ! K ,resp.) is an isomorphism.Proof. By Remark 3.2, the map (32) is an isomorphism if and only if the map φ is anisomorphism. The conclusion follows from the fact that since i is a closed embedding, i ∗ isfully faithful. The second assertion is proved using the same construction, with the arrowsreversed. Remark 3.8
If in the set-up of diagram (27) the maps J : Y → Y and i : H → Y arearbitrary locally closed embedding of varieties, then the proof of Lemma 3.7 shows that if the base change map i ∗ J ∗ K → J ∗ i ∗ K is an isomorphism, then the map (32) is also anisomorphism . In this section, Y is a quasi projective variety equipped with a fixed affine embedding Y ⊆ P N is some projective space. Let us stress that we shall consider hyperplane sectionswith respect to this fixed affine embedding.If Y is affine, then every embedding into projective space is affine. Not every embeddingof a quasi projective variety is affine, e.g. A \ { (0 , } ⊆ P . Affine embeddings alwaysexist: take an arbitrary embedding (with associated closure) Y ⊆ b Y ⊆ P M into someprojective space and blow up the boundary b Y \ Y ; the resulting projective variety Y contains Y and the complement is a Cartier divisor, so that Y ⊆ Y is an affine embedding;finally embed Y into some projective space P N : this embedding is affine. If the embeddingis not chosen to be affine, then the conclusion of Theorem 3.10 is false, as it is illustratedby the example of the punctured plane.We need the following standard vanishing result due to M. Artin. Theorem 3.9
Let Y be an affine variety and Q ∈ P Y be a perverse sheaf on Y . Then H r ( Y, Q ) = 0 , ∀ r > , H rc ( Y, Q ) = 0 , ∀ r < . Proof.
See [2] and [4]. 20et Λ ⊆ P N be a hyperplane, H := Y ∩ Λ ⊆ Y be the corresponding hyperplane sectionand consider the corresponding open and closed immersions. H i −→ Y j ←− U := Y \ H. The following is Beilinson’s version of the Lefschetz Hyperplane Theorem. The proof isan application of the base change Proposition 3.4. One can also invoke (as in [3], Lemma3.3) the generic base change theorem and reach the same conclusion (without specifyinghow one should choose the hyperplane).
Theorem 3.10
Let Q ∈ P Y be a perverse sheaf on Y . If Λ is a general hyperplane (forthe given affine embedding Y ⊆ P N ), then H r ( Y, j ! j ! Q ) = 0 , ∀ r < , H rc ( Y, j ∗ j ∗ Q ) = 0 , ∀ r > . Moreover, if Y is affine, then H r ( Y, j ! j ! Q ) = 0 , ∀ r = 0 , H rc ( Y, j ∗ j ∗ Q ) = 0 , ∀ r = 0 . Proof.
The idea of proof is to identify the cohomology groups in question (cohomologygroups with compact supports, resp.) with cohomology groups with compact supports(cohomology groups , resp.) on an auxiliary affine variety, and then apply Artin vanishingTheorem 3.9.Let Y ⊆ P N be the closure of Y . We have the following chain of equalities (see (27)) H r ( Y, j ! j ∗ Q ) = H r ( Y , J ∗ j ! j ∗ Q ) = ←− H r ( Y , j ! J ∗ j ∗ Q ) = H rc ( Y , j ! J ∗ j ! Q ) = H rc ( U , J ∗ j ! Q ) , where we have applied Lemma 3.7 and Proposition 3.4 (applied to i, J ) to obtain thesecond equality. Since j is an open immersion, j ! = j ∗ and j ! Q is perverse. Since J is an affine open immersion, J ∗ j ! Q is perverse. Since U is affine, the last group is zero for r > H c ( Y, j ∗ j ∗ Q ), is proved in a similar way. The relevant sequence ofidentifications and maps is H c ( Y, j ∗ j ∗ K ) = H c ( Y , J ! j ∗ j ∗ K ) = −→ H c ( Y , j ∗ J ! j ∗ K ) = H ( Y , j ∗ j ∗ J ! K ) = H ( U , j ∗ J ! K ) . In this section Y is a quasi projective variety and we fix an affine embedding Y ⊆ P N .Let Λ , Λ ′ ⊆ P N be two hyperplanes, H := Y ∩ Λ ⊆ Y and j : Y \ H := U → Y ← H : i be the corresponding open and closed immersions. Note that j ! = j ∗ . Similarly, we haveΛ ′ , H ′ , U ′ , i ′ , j ′ . We have the Cartesian diagram of open embeddings U j (cid:15) (cid:15) U ∩ U ′ j ′ o o j (cid:15) (cid:15) Y U ′ . j ′ o o (33)21ince the embedding Y ⊆ P N is affine, these open embeddings are affine and so are theopen sets U, U ′ , U ∩ U ′ . If the embedding were not affine, these open sets may fail to beaffine and the conclusions on vanishing of Theorem 3.12 would not hold. Using the naturalmaps and isomorphisms (19, 20, 21) and that j ! = j ∗ , j ′ ! = j ′∗ , we get the following maps j ! j ! j ′∗ j ′∗ = −→ j ! j ′∗ j ! j ′∗ = −→ j ! j ′∗ j ′∗ j ! c ′ −→ j ′∗ j ! j ′∗ j ! = −→ j ′∗ j ′∗ j ! j ! whose composition we denote by c : j ! j ! j ′∗ j ′∗ −→ j ′∗ j ′∗ j ! j ! . (34)The octahedron axiom applied to the composition j ′∗ j ′∗ −→ i ∗ i ∗ j ′∗ j ′∗ ψ −→ i ∗ j ′∗ i ∗ j ′∗ (= j ′∗ i ∗ i ∗ j ′∗ ) (35)yields a distinguished triangle j ! j ! j ′∗ j ′∗ [1](= j ! j ′∗ j ! j ′∗ [1]) −→ j ′∗ j ! j ! j ′∗ [1] −→ Cone( ψ ) −→ (36) Lemma 3.11
The map c : j ! j ! j ′∗ j ′∗ −→ j ′∗ j ′∗ j ! j ! is an isomorphism if and only if the base change map i ∗ j ′∗ j ′∗ −→ j ′∗ i ∗ j ′∗ is an isomorphism.Proof. Same as Lemma 3.7.
Theorem 3.12
Let Q ∈ P Y . If (Λ , Λ ′ ) is a general pair, then we have j ! j ! j ′∗ j ′∗ Q = j ′∗ j ′∗ j ! j ! Q and H r ( Y, j ! j ! j ′∗ j ′∗ Q ) = H rc ( Y, j ′∗ j ′∗ j ! j ! Q ) = 0 , ∀ r = 0 . Proof.
For a fixed and arbitrary Λ ′ , by virtue of Lemma 3.11 and Proposition 3.4 (appliedto i, j ′ ), the first equality holds for Λ general. This implies that the first equality holdsfor a general pair.We prove the statement in cohomology. The one in cohomology with compact supportsis proved in a similar way, by switching the roles of the two hyperplanes Λ and Λ ′ . Notethat j ′∗ j ′∗ Q is perverse. The vanishing of the groups for r < r > H r ( Y, j ! j ! j ′∗ j ′∗ Q ) = H r ( Y, j ′∗ j ′∗ j ! j ! Q ) = H r ( U ′ , j ′∗ j ! j ! Q ); U ′ is affine, j ′∗ j ! j ! Q is perverse and the last group iszero for r < Remark 3.13
Theorem 3.12 is due to Beilinson [3] and it is used in [8] to describe perversefiltrations on quasi projective varieties using general pairs of flags.22 .3.4 The Lefschetz hyperplane theorem for constructible sheaves
As it is observed in [17], Introduction, Theorem 3.10 admits a sheaf-theoretic versionwhich we state and prove below. Let Y ⊆ P N be a quasi projective variety of dimension n embedded in some projective space in such a way that the embedding is affine. Let V ⊆ P N be a hypersurface, V := Y ∩ V and j : Y \ V → Y . Theorem 3.14
Let T be a constructible sheaf on Y . There is a hypersurface V such that1. H r ( Y, j ! j ! T ) = 0 , for every r < n (for every r = n if Y is affine),2. dim V = dim Y − .Proof. Let Σ be a stratification of Y with respect to which T is constructible. The union S n of all n -dimensional strata is a non-empty, Zariski open subvariety of Y with the propertythat T | S n is locally constant and dim ( Y \ S n ) ≤ n −
1. Let V ′ ⊆ P N be a hypersurfacecontaining Y \ S n but not containing any of the irreducible components of S n . Since theopen embedding j ′ : Y \ V ′ → Y is affine, j ′ ! j ′ ! T [ n ] is a perverse sheaf on Y . We applyTheorem 3.10 to this perverse sheaf and conclude that the desired hypersurface is of theform V := V ′ ∪ Λ for some general hyperplane Λ.
Remark 3.15
The hypersurface V ′ must contain the “bad locus” of the sheaf T . Inparticular, it is a “special” hypersurface of sufficiently high degree. As the proof shows, itis not necessary to achieve 2. in order to achieve 1. However, 2. is useful in procedureswhere one uses induction on the dimension. I do not know of a version of Theorem 3.14for cohomology groups with compact supports. The Generic Base Change Theorem was proved in [10] as an essential ingredient, in the´etale context, towards the constructibility for direct images of complexes with constructiblecohomology sheaves for morphisms of finite type over a field. These kinds of constructibil-ity results are fundamental and permeate the whole theory of ´etale cohomology.In this section we want to state the Generic Base Change Theorem and show how it canbe applied in practice when one has a base change issues with “parameters,” e.g. elementsof a linear system. For example, in the proof of the Lefschetz hyperplane theorem 3.10 ifone can afford to work with general linear sections, then Proposition 3.21 can be used inplace of Proposition 3.4.
Let X f → Y p → S and S ′ → S be maps. Denote by X ′ f → Y ′ p → S ′ the varieties and mapsobtained by base change via the given S ′ → S. C ∈ D X . One says that the formation of f ∗ C commutes with arbitrary base change if, for every S ′ → S, the resulting first base change map (20) g ∗ f ∗ C → f ∗ g ∗ C is anisomorphism.The issue does not arise for f ! , in fact, the base change isomorphism (21) g ∗ f ! = f ! g ∗ implies that for every C ∈ D X the formation of f ! C commutes with arbitrary base change.Given a stratification Σ of X and a map f : X → Y, one can refine Σ so that therefinement is part of a stratification of the map f. It follows at once that, given f : X → Y and C ∈ D X , there is an Zariski-dense open subset U ⊆ Y with the property that, given f − ( U ) → U = U, the formation of f ∗ ( C | f − U ) commutes with arbitrary base change. Itis sufficient to take for U the dense open stratum on Y of the stratification for f refiningthe one for C . In fact, f is then topologically locally trivial over U and the base changemaps are then isomorphisms.However, what above is insufficient to prove the, for example, the vanishing Theorem3.10. Moreover, it cannot be used for example to work with constructible sheaves for the´etale topology for varieties over a field, where one cannot achieve the local triviality of f : X → Y over U ⊆ Y (in fact, the generic base change theorem is a tool that effectivelyfixes this problem at the level of sheaves).The Generic Base Change Theorem is a tool apt to deal with these and other situations. Theorem 3.16
Let X f → Y p → S be maps and C ∈ D X . There exists a Zariski open anddense subset V ⊆ S such that, if one takes ( pf ) − ( V ) → p − V → V, then the formationof f ∗ ( C | ( pf ) − V ) commutes with arbitrary base change T → V. Proof.
For the ´etale case see [10], [Th. finitude], Th. 1.9. The proof in the case of complexvarieties and C ∈ D X is similar and, in fact, simpler. Remark 3.17
Note that the open set V depends on C . However, an inspection of theproof reveals that given a stratification Σ of X , one can choose the Zariski open and densesubset V ⊆ S so that the conclusion of the Generic Base Change Theorem holds for every C ′ ∈ D Σ X . Remark 3.18
For Σ and V as above, the formation of f ! commutes with the formationof g ! over V for every K ∈ D Σ X . In fact, to prove that g ! f ! K ← f ! g ! K is an isomorphismfor K ∈ D Σ X , it is sufficient to observe that K ∨ ∈ D Σ X and dualize the isomorphism g ∗ f ∗ K ∨ → f ∗ g ∗ K ∨ which holds over V by Theorem 3.16. The following standard lemma is an illustration of the use of Generic Base Change. Itis essentially a special case, formulated in a way that directly applies to the situationdealt-with in the Lefschetz Hyperplane Theorem.24 emma 3.19
Let f : X → Y be a map, C ∈ D X and X ,T τ ′ / / f ,T (cid:15) (cid:15) i ′ T ( ( X ,Vf ,V (cid:15) (cid:15) u ′ / / X f (cid:15) (cid:15) v ′ / / g ′ X f (cid:15) (cid:15) π ′ / / X f (cid:15) (cid:15) Y ,Tp T (cid:15) (cid:15) τ / / i T Y ,Vp V (cid:15) (cid:15) u / / Y p (cid:15) (cid:15) v / / g > > Y π / / YT t / / V u ′′ / / S be a commutative diagram with Cartesian squares satisfying:1. π smooth; in particular, π ∗ f ∗ ≃ f ∗ π ′∗ ;2. g smooth; in particular, g ∗ f ∗ ≃ f ∗ g ′∗ ;3. V ⊆ S is a Zariski-dense open subset such that the formation of f ,V ∗ u ′∗ v ′∗ π ′∗ C commutes with arbitrary base change (on V ).For every t : T → V, the natural base change map is an isomorphism: i ∗ T f ∗ C ≃ −→ f ,T ∗ i ′∗ T C. Proof.
The natural map i ∗ T f ∗ C −→ f ,T ∗ i ∗ T C factors as follows i ∗ T f ∗ C = τ ∗ u ∗ g ∗ f ∗ C −→ τ ∗ u ∗ f ∗ g ′∗ C −→ τ ∗ f ,V ∗ u ′∗ g ′∗ C −→ f ,T ∗ τ ′∗ u ′∗ g ′∗ C = f ,T ∗ i ′∗ T C. Since g and u are smooth, the first and second arrows are isomorphisms The third one isan isomorphism by the choice of V. Remark 3.20
Fix a stratification Σ for X . As in Remark 3.17, we can choose V so that3. above holds for every C ′ ∈ D Σ X and conclude (see Remark 3.18) that we have the basechange isomorphisms i ∗ T f ∗ C ′ ≃ −→ f ,T ∗ i ′∗ T C ′ , i ! T f ! C ′ ≃ ←− f ,T ! i ′ T ! C ′ , ∀ T −→ V, ∀ C ′ ∈ D Σ Y . We now apply Lemma 3.19 and Remark 3.20 to the following situation: let f : X → Y be a map of varieties, | H | be a finite dimensional and base-point-free linear system on Y , e.g. the very ample linear system associated with an embedding on Y into projectivespace. Given H ∈ | H | , we have the Cartesian diagram (24).25 roposition 3.21 Let Σ be a stratification of X . If H ∈ | H | is general, then for every C ∈ D Σ X the base change maps i ∗ f ∗ C −→ f ∗ i ∗ C, i ! f ! C ←− f ! i ! C are isomorphisms.1st proof (it uses the generic base change theorem and it does not single out a specific H ). We only need to apply Lemma 3.19 and Remark 3.20 to the following situation: Y := Y × | H | , Y ⊆ Y × | H | the universal hyperplane section, S := | H | and t : T → V isthe embedding of a closed point. (it uses Proposition 3.4 and it identifies precisely which conditions on H mustbe satisfied). A general hyperplane Λ is transverse to all the strata of a fixed stratificationof f ∗ C . This means that i : H → Y is a normally nonsingular inclusion with respect to thegiven stratification. For such a Λ, i ∗ f ∗ C = i ! [2] f ∗ C = f ∗ i ! C [2]. For Λ general, i : X H → X is transverse to all the strata of a fixed stratification of C and we have i ! C = i ∗ C [ − References [1] D. Arapura, “The Leray spectral sequence is motivic,” Inv. Math. (2005), no.3567-589.[2] M. Artin, “Th´eor`eme de finitude pour un morphisme propre; dimension coho-mologique des sch´emas alg´ebriques affines,” SGA 4, Lecture Notes in Math. 305(1973).[3] A.A. Beilinson, “On the derived category of perverse sheaves,” K -theory, arith-metic and geometry (Moscow, 1984–1986) , pp.27–41, Lecture Notes in Math., 1289,Springer 1987.[4] A.A. Beilinson, J.N. Bernstein, P. Deligne, Faisceaux pervers , Ast´erisque , Paris,Soc. Math. Fr. 1982.[5] A. Borel et al.,
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