Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings
aa r X i v : . [ m a t h . AG ] J un DEPENDENCE OF LYUBEZNIK NUMBERS OF CONES OFPROJECTIVE SCHEMES ON PROJECTIVE EMBEDDINGS
THOMAS REICHELT, MORIHIKO SAITO, AND ULI WALTHER
Abstract.
We construct complex projective schemes with Lyubeznik numbers of theircones depending on the choices of projective embeddings. This answers a question ofG. Lyubeznik in the characteristic 0 case. Note that the situation is quite different inthe positive characteristic case using the Frobenius endomorphism. Reducibility of schemesis essential in our argument, and the question is still open in the irreducible singular case.
Introduction
Let X be a projective scheme over C with L a very ample line bundle. Let C be the coneof X associated with L . Let x , . . . , x n be projective coordinates of Y := P n − C containing X so that O P n − (1) | X = L . Let I be the ideal of R := C [ x , . . . , x n ] defining the cone C ⊂ A n C .The Lyubeznik numbers λ k,j ( C ) are defined by(1) λ k,j ( C ) := dim C Ext kR ( C , H n − jI R ) ( k, j ∈ N ) , see [Ly1, Ly2, NWZ]. Here the H n − jI R are the local cohomology groups, and vanish for j > dim C , see Remark (ii) after (1.1) below. The higher extension groups Ext kR ( C , ∗ ) canbe calculated by the Koszul complex for the multiplications of the x i ( i ∈ [1 , n ]) which givesa free resolution of C over R . This holds also for the higher torsion groups Tor Rn − k ( C , ∗ ).Setting V := Spec R = A n C , we then get the isomorphisms(2) Ext kR ( C , H n − jI R ) = Tor Rn − k ( C , H n − jI R ) = H k − n L i ∗ ,V ( H n − jC O V ) . Here i A,B : A ֒ → B denotes an inclusion of a subset A ⊂ B in general, L i ∗ ,V means thederived pull-back functor for O -modules endowed with an integrable connection (that is, forleft D -modules), and the H n − jC are the algebraic local cohomology functors for the closedsubscheme C ⊂ V . Note that O V is algebraic so that Γ( V, O V ) = R .Using the Riemann-Hilbert correspondence, we then get the following (see (1.1) below). Proposition 1.
In the above notation, we have the equalities λ k,j ( C ) = dim Q H k i !0 ,C ( p H − j DQ C ) ( k, j ∈ N ) . (See also [GS].) Here Q C an and its dual DQ C an (see [Ve1]) are respectively denoted by Q C and DQ C to simplify the notation (where C an is the analytic space associated with C ),and similarly for Q X , DQ X . The p H j are the cohomology functors associated with thetruncations p τ k constructed in [BBD] (see also [Di, KS]). Note that the usual cohomologyfunctors H j for bounded complexes of D -modules having regular holonomic cohomologysheaves correspond to the functors p H j by the Riemann-Hilbert correspondence.Setting(3) F j := p H − j DQ C , F ′ j := F j | C ′ with C ′ := C \ { } ( j ∈ N ) , we have the following (see (1.2) below). Proposition 2.
For k > , there are isomorphisms H k i !0 ,C F j = H k − i ∗ ,C R ( i C ′ ,C ) ∗ F ′ j = H k − ( C ′ , F ′ j ) . Combined with Propositions 1, this implies the following.
Corollary 1.
We have λ k,j ( C ) = dim Q H k − ( C ′ , F ′ j ) ( k > . For k ∈ Z , j ∈ N , set(4) H k ( j ) ( X ) := H k ( X, p H − j DQ X ) ,H k ( j ) ( X ) L := Ker (cid:0) c ( L ) : H k ( j ) ( X ) → H k +2( j ) ( X )(1) (cid:1) ,H k ( j ) ( X ) L := Coker (cid:0) c ( L ) : H k − j ) ( X )( − → H k ( j ) ( X ) (cid:1) , where ( m ) denotes a Tate twist for m ∈ Z , see [De1]. (Note that a subquotient of H k ( j ) ( X )is identified with Gr jG H j − k ( X ) by a spectral sequence, where { G j } is a decreasing filtrationon H j − k ( X ) = H k − j ( X, DQ X ) induced by p τ − j on DQ X , see [BBD, Ve2].)Using a generalized Thom-Gysin sequence, we get the following (see (1.3–4) below). Proposition 3.
There are short exact sequences → H k ( j − ( X ) L (1) → H k − ( C ′ , F ′ j ) → H k − j − ( X ) L → k ∈ Z ) . Combining this with Corollary 1, we get the following.
Corollary 2.
The Lyubeznik numbers λ k,j ( C ) of the cone C of a projective scheme X dependon the choice of a very ample line bundle L if the following condition holds :(5) dim H k ( j − ( X ) L + dim H k − j − ( X ) L depends on L for some k > j > Q -homologymanifold case (generalizing [Swl] in the non-singular case), see Corollary 1.8 below. UsingCorollary 2, we can prove the following. Theorem 1.
There are projective schemes over C such that their irreducible componentsare smooth and the Lyubeznik numbers λ k,j ( C ) of their cones C depend on the choices ofprojective embeddings for some k > . Here j coincides with the dimension of the lowestdimensional irreducible component of C , and X can be equidimensional. This answers a question of G. Lyubeznik [Ly2] in the characteristic 0 case, see [NWZ, Zh]for the positive characteristic case and [Swl] for the X non-singular case. Theorem 1 may berather unexpected, since the situation is entirely different in the positive characteristic casewhere the Frobenius endomorphism can be used. In the case of schemes over C , the proofof Theorem 1 is reduced by Corollary 2 to finding complex projective schemes X satisfyingcondition (5). It is rather easy to construct such schemes if the condition k > k may be negative), see (2.1) below. To satisfy the last condition, we need some moreconstruction, where the argument is much easier in the non-equidimensional case (see (2.2)below), and we have to use Hodge theory in the equidimensional case (see (2.3) below). Inthese arguments, reducibility of schemes is essential, and the question is still open in the X irreducible singular case.The first named author was supported by a DFG Emmy-Noether-Fellowship (RE 3567/1-1) and acknowledges partial support by the project SISYPH: ANR-13-IS01-0001-01/02 andDFG grant HE 2287/4-1 & SE 1114/5-1. He would like to thank Duco van Straten for a stim-ulating discussion. The second named author is partially supported by Kakenhi 15K04816.The third named author is supported in part by NFS grant DMS-1401392 and by SimonsFoundation Collaboration Grant for Mathematicians EPENDENCE OF LYUBEZNIK NUMBERS 3
In Section 1 we review generalized Thom-Gysin sequences, study the Lyubeznik numbersin the X Q -homology manifold case, and prove Propositions 1–3. In Section 2 we proveTheorem 1 by constructing desired examples.
1. Preliminaries
In this section we review generalized Thom-Gysin sequences, study the Lyubeznik numbersin the X Q -homology manifold case, and prove Propositions 1–3. By the Riemann-Hilbert correspondence for algebraic D -modules using the de Rham functor DR (see for instance [Bo]), the derived pull-back functor L i ∗ ,V [ − n ] (explained before Proposition 1) corresponds to i !0 ,V , that is,(1 . .
1) DR ◦ L i ∗ ,V [ − n ] = i !0 ,V ◦ DR , see also Remark (iii) below (and [Sa2, Remark after Corollary 2.24]). We have moreover(1 . .
2) DR( O V [ n ]) = DC V (= C V [2 n ]) , since DR( O V ) = C V [ n ]. Here D denotes the dual functor, see [Ve1]. Note that the functor L i ∗ ,V [ − n ] corresponds to i ∗ ,V under the contravariant functor Sol = D ◦ DR, see Remark (i)below and also [Ka], [KK], [Me], [Sa1, Remark 2.4.15 (3)]. (The equivalence of categoriesitself is not really needed here.)The derived local cohomology functor R Γ C corresponds to ( i C,V ) ∗ i ! C,V (see [Bo]), and wehave(1 . . i ! C,V DQ V = D i ∗ C,V Q V = DQ C . So Proposition 1 follows.
Remarks. (i) Let X be a complex manifold (or a smooth complex algebraic variety) ofdimension n . For a bounded complex M • of left D X -modules having regular holonomiccohomology sheaves, the de Rham and solution functors can be defined by(1 . .
4) DR( M • ) := R H om D X ( O X , M • )[ n ] , Sol( M • ) := R H om D X ( M • , O X )[ n ] . (If X is a smooth complex algebraic variety, X and M • on the right-hand side are respectivelyreplaced by X an and M an • .) Taking the composition, we can get a perfect pairing(1 . .
5) DR( M • ) ⊗ C Sol( M • ) → R H om D X ( O X , O X )[2 n ] = C X [2 n ] = DC X , that is, the corresponding morphism(1 . .
6) DR( M • ) → D (cid:0) Sol( M • ) (cid:1) := R H om C X (cid:0) Sol( M • ) , DC X (cid:1) is an isomorphism. It is also known that DR commutes with D . (This follows, for instance,from [Sa1, Proposition 2.4.12].)(ii) In the notation of the introduction, we have by [Ha, Theorem 3.8] (see also [BS, Iy])(1 . . H n − jI R = 0 for n − j < codim C. In our case this follows by taking a minimal dimensional complete intersection containing C ⊂ C n and using the composition of derived local cohomology functors. Note also that, bythe Riemann-Hilbert correspondences and (1.1.2), this vanishing is equivalent to(1 . . p H − j R Γ C DQ X = 0 (that is, p H j Q C = 0 ) for j > dim C, and the assertion for Q C follows easily from the definition of the t -structure in [BBD]. T. REICHELT, M. SAITO, AND U. WALTHER (iii) Let X be a closed submanifold of a complex manifold Y . For a regular holonomicright D Y -module M , there are natural inclusions (as O Y -modules)(1 . . E xt j O Y ( O X , M ) ֒ → H j [ X ] M ( j ∈ N ) , inducing isomorphisms of right D Y -modules(1 . . E xt j O Y ( O X , M ) ⊗ D X D X → Y ∼ −→ H j [ X ] M ( j ∈ N ) . Here D X֒ → Y := O X ⊗ O Y D Y , and the H j [ X ] M are the algebraic local cohomology sheavesdefined by(1 . . H j [ X ] M := k → lim E xt j O Y ( O Y / I kX , M ) ( j ∈ N ) , with I X ⊂ O Y the ideal sheaf of X ⊂ Y , see [KK]. Note that the sources of (1.1.9) and(1.1.10) are respectively the cohomological pull-back of M as right D -module by the inclusion i X,Y : X ֒ → Y and its direct image as right D -module by i X,Y . (Using a spectral sequence,the proof of (1.1.10) can be reduced to the codimension 1 case.)Set r := codim Y X . The formula corresponding to (1.1.10) for a regular holonomic left D -module M is as follows (see, for instance, the proof of [KK, Corollary 5.4.6]):(1 . . D Y ← X ⊗ D X T or O Y r − j ( O X , M ) = H j [ X ] M ( j ∈ N ) . The last isomorphism in Proposition 2 holds, since 0 is thevertex of the cone C . The first isomorphism follows from the long exact sequence associatedwith the distinguished triangle(1 . . i !0 ,C F j → i ∗ ,C F j → i ∗ ,C R ( i C ′ ,C ) ∗ i ∗ C ′ ,C F j +1 → , since H k i ∗ ,C F j = 0 for k >
0, see [BBD] (and also [Sa2, Remark after Corollary 2.24]). Thelast triangle can be obtained by applying the functor i ∗ ,C to the triangle( i ,C ) ∗ i !0 ,C → id → R ( i C ′ ,C ) ∗ i ∗ C ′ ,C +1 → . This finishes the proof of Proposition 2. (see also [Ko, Swz]). Let π : E → X be avector bundle of rank r on a complex analytic space X which is assumed connected. Set E ′ := E \ X , where X is identified with the zero section of E . There are natural morphisms i X : X ֒ → E, j E ′ : E ′ ֒ → E, π ′ := π | E ′ : E ′ → X. For F • ∈ D bc ( X, Q ), we have the distinguished triangle(1 . . F • ξ → F • ( r )[2 r ] → R π ′∗ π ′ ! F • +1 → , inducing a long exact sequence called a generalized Thom-Gysin sequence :(1 . . → H k ( X, F • ) → H k +2 r ( X, F • )( r ) → H k ( E ′ , π ′ ! F • ) → H k +1 ( X, F • ) → . Indeed, the triangle (1.3.1) is identified with the distinguished triangle(1 . . i ! X π ! F • → R π ∗ π ! F • → R π ′∗ π ′ ! F • +1 → , since π ! F • = π − F • ( r )[2 r ]. The last triangle is obtained by applying R π ∗ to(1 . .
4) ( i X ) ∗ i ! X π ! F • → π ! F • → R ( j E ′ ) ∗ j ∗ E ′ π ! F • +1 → . The morphism ξ in (1.3.1) is induced by the Euler class of E via the adjunction isomorphismfor a ∗ X and ( a X ) ∗ :(1 . .
5) Hom( Q X , Q X ( r )[2 r ]) = Hom (cid:0) Q , R Γ( Q X ( r )[2 r ]) (cid:1) = H r ( X, Q )( r ) . EPENDENCE OF LYUBEZNIK NUMBERS 5
Here Hom denotes the group of morphisms in the derived categories, and the
Euler class of E is the image of 1 by the following morphism which is induced by ξ for F • = Q X :(1 . . Q = H ( X, Q ) → H r ( X, Q )( r ) , see also [KS, Ex. III.7]. Note that the Euler class of E (denoted by e ) induces the morphism ξ in (1.3.1) by using its tensor product with the identity on F • as follows:(1 . . F • = Q X ⊗ Q F • e ⊗ id −→ Q X ( r )[2 r ] ⊗ Q F • = F • ( r )[2 r ] . Indeed, ξ coincides with the tensor product of ξ for F • = Q X (that is, the Euler class of E )with the identity on F • . This is shown by taking π ∗ of the commutative diagram(1 . .
8) (Γ X I • ) ⊗ Q π − F • ֒ → I • ⊗ Q π − F • ↓ ↓ Γ X J • ֒ → J • where I • is a flasque resolution of Q E = π − Q X , and J • is a flasque resolution of I • ⊗ π − F • . Remark.
The long exact sequence (1.3.2) holds in the category of mixed Q -Hodge structuresif F • underlies a bounded complex of mixed Hodge modules. Indeed, the above constructioncan be lifted naturally in the category of mixed Hodge modules, see [Sa2]. In the notation of (1.3) and the introduction, we have(1 . . π ′ ! F • = F ′ j • [1] by setting F • := p H − j +1 DQ X , where E ′ = C ′ and r = 1. So Proposition 3 follows from (1.3.2). Q -homology manifold case (see [GS] for the X smooth case). Let X be a projectivescheme such that(1 . . p H j Q X = 0 ( j = d ) , where d ∈ Z > . This condition is satisfied for instance if X an is purely d -dimensional, andis a Q -homology manifold or analytic-locally a complete intersection. (The proof of the lastassertion follows, for instance, by using the Riemann-Hilbert correspondence and the localcohomology sheaves defined as in (1.1.11).)In the notation of the introduction, the assumption (1.5.1) implies that(1 . .
2) Supp p H k DQ C , Supp p H k R ( j C ′ ) ∗ DQ C ′ ⊂ { } ( k = − d − . We have the distinguished triangle(1 . . Q { } → DQ C → R ( j C ′ ) ∗ DQ C ′ +1 → , which is the dual of the short exact sequence0 → ( j C ′ ) ! Q C ′ → Q C → Q { } → . In this section j C ′ ,C and i ,C are denoted respectively by j C ′ and i to simplify the notation.We have the Leray-type spectral sequences(1 . . ∗ E p,q = H p i ∗ p H q R ( j C ′ ) ∗ DQ C ′ = ⇒ H p + q i ∗ R ( j C ′ ) ∗ DQ C ′ , (1 . . ! E p,q = H p i !0 p H q R ( j C ′ ) ∗ DQ C ′ = ⇒ H p + q i !0 R ( j C ′ ) ∗ DQ C ′ = 0 , where the last vanishing follows from i !0 R ( j C ′ ) ∗ = 0 (the latter is the dual of i ∗ R ( j C ′ ) ! = 0).These can be constructed by using spectral objects in [Ve2] together with an argumentsimilar to [De1, Example 1.4.8], or we can use the Riemann-Hilbert correspondence after thescalar extension by Q ֒ → C . T. REICHELT, M. SAITO, AND U. WALTHER
By (1.1.8), (1.5.2) together with properties of i ∗ , i !0 in [BBD] (see also [Sa2, Remark afterCorollary 2.24]), we get(1 . . ∗ E − p,q = ! E p,q = 0 unless p = 0 , q > − d − q = − d − , p > . The generalized Thom-Gysin sequence (1.3.2) together with an isomorphism similar to thelast isomorphism of Proposition 2 implies that(1 . . H − k i ∗ R ( j C ′ ) ∗ DQ C ′ = 0 unless k ∈ [1 , d + 2] . (This can be shown also by using the link L C, of C at 0. It is the intersection of C with asphere S n − around 0 ∈ C n , and is a (2 d +1)-dimensional real analytic space, see [DS]. Itsdualizing complex DQ L C, (see [Ve1]) is isomorphic to the restriction of DQ C ′ [ −
1] to L C, .)The spectral sequence (1.5.4) degenerates at E by (1.5.6), and it follows from (1.1.8),(1.5.2), (1.5.7) that(1 . . p H − j R ( j C ′ ) ∗ DQ C ′ = 0 unless j ∈ [1 , d + 1] . Using (1.5.3) and (1.5.8), we can prove the isomorphisms(1 . . p H k DQ C ∼ −→ p H k R ( j C ′ ) ∗ DQ C ′ ( k = − , together with the short exact sequence(1 . .
10) 0 → p H − DQ C → p H − R ( j C ′ ) ∗ DQ C ′ → Q { } → . Here the vanishing of p H DQ C is rather nontrivial. If we have p H DQ C = 0, then (1.5.8)and the long exact sequence associated with (1.5.3) imply the surjectivity of the composition Q { } → DQ C → p H DQ C . We then get a splitting of the first morphism, but this is a contradiction. So the vanishingof p H DQ C follows.Combined with (1.5.6), the spectral sequence (1.5.5) implies the isomorphisms(1 . . H p i !0 p H − d − R ( j C ′ ) ∗ DQ C ′ = ( p H p − d − R ( j C ′ ) ∗ DQ C ′ if p > , p = 0 , . Here the direct image ( i ) ∗ is omitted on the left-hand side to simplify the notation.By (1.5.9–11), we get the following (see [GS] for the X smooth case). Under the assumption (1 . . , we have (1 . . λ k,j ( C ) = 0 unless j = d + 1 , k ∈ [2 , d + 1] or k = 0 , j ∈ [1 , d ] , and moreover the following relations among the Lyubeznik numbers hold :(1 . . λ k,d +1 ( C ) = λ ,d +2 − k ( C ) + δ k,d +1 ( k ∈ [2 , d + 1]) , where δ k,d +1 = 1 if k = d + 1 , and otherwise. This implies the following.
Under the assumption (1 . . , the converse of Corollary holds. We then get the following generalization of [Swl] in the X non-singular case. The Lyubeznik numbers λ k,j ( C ) of the cone C of a complex projectivescheme X are independent of the choice of a projective embedding of X , if the associatedanalytic space X an is a Q -homology manifold. EPENDENCE OF LYUBEZNIK NUMBERS 7
Proof.
By the definition of Q -homology manifold, we have H j { x } Q X = Q if j = 2 dim X ,and 0 otherwise ( ∀ x ∈ X an ). By induction on strata, we see that the composition of thefollowing two canonical morphisms is an isomorphism (see [GM, BBD]):(1 . . Q X [dim X ] → IC X Q → DQ X ( − dim X )[ − dim X ] , where the last morphism is the dual of the first. This implies that p H j ( Q X [dim X ]) = 0( j = 0), and the above two morphisms are both isomorphisms by [GM] or using the simplicityof the intersection complex IC X Q . Moreover the hard Lefschetz theorem holds for theintersection cohomology of the complex projective variety X , see [BBD] (and [Sa1]). Thedimensions of the kernel and cokernel of the action of c ( L ) on the (intersection) cohomologyare then read off from the Betti numbers of X by using the primitive decomposition. So theassertion follows from Corollary 1.7. This finishes the proof of Corollary 1.8. Remark.
The independence of the Lyubeznik numbers of cones also holds if X has onlyisolated singularities and (1.5.1) is satisfied.
2. Construction of examples
In this section we prove Theorem 1 by constructing desired examples. k ∈ Z ”. Let Y be a smoothcomplex projective variety of dimension d > very ample divisors D , D ′ suchthat their Chern classes c ( D ), c ( D ′ ) are linearly independent. (The last condition can besatisfied in case the Picard number of Y is at least 2, for instance, if Y is P × P or a onepoint blow-up of P .) Consider the line bundle L (as a variety) corresponding to D . (Inthis paper a line bundle usually means an invertible sheaf .) We have the section Y of L corresponding to D . Here we may assume D is effective, reduced and smooth. Then Y intersects the zero section Y of L transversally along D . Set X := Y ∪ Y ⊂ L . This is adivisor with normal crossing, and we get(2 . . p H d Q X = Q X [ d ] , p H − d DQ X = DQ X [ − d ] . Using the dual of the short exact sequence0 → Q X → Q Y ⊕ Q Y → Q D → , which is identified with a distinguished triangle, we get the long exact sequence(2 . . → H j − ( D )( − i ∗ → H j ( Y ) ⊕ H j ( Y ) → H j − d (cid:0) X , DQ X ( − d ) (cid:1) → H j − ( D )( − → · · · , inducing the isomorphisms(2 . .
3) Gr W H − d (cid:0) X , DQ X ( − d ) (cid:1) = Coker (cid:0) H ( D )( − i ∗ ֒ → H ( Y ) ⊕ H ( Y ) (cid:1) ,H − d (cid:0) X , DQ X ( − d ) (cid:1) = H ( Y ) ⊕ H ( Y ) . Here i ∗ denotes the Gysin morphism for the inclusion i : D ֒ → Y , and dim H ( D ) = 1 bythe weak Lefschetz theorem. Since X is finite over Y , and a finite morphism is ample inthe sense of Grothendieck with relatively ample line bundle trivial, the ample line bundlescorresponding to D, D ′ on Y give ample line bundles on X via the pull-back by the naturalmorphism π Y : X → Y , see [Gr, Propositions 4.4.10 and Corollary 6.1.11] and Remark (i)below.By (2.1.3) (together with (2.1.1)), we get a difference in the dimension of the images of(2 . . c ( D ) , c ( D ′ ) : H − d ( X , p H − d DQ X )( − → H − d ( X , p H − d DQ X ) . T. REICHELT, M. SAITO, AND U. WALTHER
Here the action of ℓ ∈ H ( X )(1) on F ∈ D bc ( X , Q ) can be defined by taking the tensorproduct of F with the morphism Q X → Q X (1)[2] defined by ℓ . (This is compatible withthe normalization.)As a conclusion, condition (5) is satisfied (except for the condition k >
2) with(2 . . j − d , k = 2 − d ( < . if we further assume the vanishing of H ( Y ) so that H k − j − ( X ) = 0.This construction will be used in (2.2–3) below, where k can be “shifted” by taking a product with an appropriate projective scheme (and using the Segre embedding) so that thecondition k > Remarks. (i) The pull-back to X of the very ample line bundle L Y := O Y ( D ) is veryample, since D is assumed smooth. Indeed, there is a short exact sequence of O X -modules0 → O X → O Y ⊕ O Y → O D → . Taking the tensor product with L := π ∗ Y L Y , we can deduce thatΓ( X , L ) = Ker (cid:0) Γ( Y, L Y ) ⊕ Γ( Y, L Y ) δ → Γ( D, L Y | D ) (cid:1) . Here δ ( s, s ′ ) := ( s − s ′ ) | D , and Y , Y are identified with Y . We will denote by σ ( s, s ′ ) theglobal section of L corresponding to ( s, s ′ ) ∈ Ker δ . Then Γ( X , L ) contains σ ( s + s ′′ , s )for s, s ′′ ∈ Γ( Y, L Y ), where s ′′ is a defining section of D . This implies an embedding of X into a projective space. Indeed, its restrictions to Y , Y are embeddings using the inclusionΓ( Y, L Y ) ∋ s σ ( s, s ) ∈ Γ( X , L ) , which corresponds to a projection of projective spaces. Here the images of Y \ D and Y \ D are separated by s ′′ in σ ( s + s ′′ , s ) (combined with the above projection). Moreover, σ ( s ′′ , /σ ( s, s ) gives a local coordinate of Y vanishing on Y ⊂ X , if s does not vanish ata given point (and similarly for σ (0 , − s ′′ ) /σ ( s, s ) with Y , Y exchanged).The pull-back of D ′ is also very ample if D ′ − D is effective and base-point-free, where s ′′ is replaced by the product of s ′′ with a section of O Y ( D ′ − D ) (which is generated by globalsections because of the last assumption).(ii) Assume Y = P × P with D ⊂ P × P the diagonal so that b ( D ) = 0. Note that D is very ample (since it is linearly equivalent to E + E with E i ( i = 1 ,
2) the pull-backof a point of P by the i the projection P × P → P ), where we use the Segre embedding P × P ֒ → P . For D ′ , we can take a E + a E for any a , a ∈ Z > with a = a ; for instance,( a , a ) = (2 , γ, γ ′ be the dimensions of the cokernels of c ( D ), c ( D ′ ) in (2.1.4) respectively. Then d = 2, b ( Y ) = 0, b ( Y ) = 2, and(2 . . b ( X ) = 2 , b ( X ) = 3 , γ = 2 , γ ′ = 1 . Let e Z be the blow-up of P d +2 along a point P ∈ P d +2 , where d >
2. This is identified with a P -bundle over P d +1 , and we have the projection ρ : e Z → P d +1 , where the target is identified with the set of lines of P d +2 passing through P . The projection ρ has the zero-section Σ given by the exceptional divisor of the blow-up. It has anothersection Σ ∞ which is disjoint from Σ , and is given by the inverse image of a hyperplane of P d +2 not containing the center of the blow-up P . Let W ⊂ P d +1 be a linear subspace ofcodimension 2. Put Z := ρ − ( W ) , Z := Σ ⊔ Σ ∞ ⊂ e Z, EPENDENCE OF LYUBEZNIK NUMBERS 9 with dim Z = d , dim Z = d + 1 . Set X := Z ∪ Z ⊂ e Z, Z ′ := Z \ Z . Let j Z ′ : Z ′ ֒ → Z be the natural inclusion. We have the short exact sequence(2 . .
1) 0 → ( j Z ′ ) ! Q Z ′ → Q X → Q Z → . Taking the dual, we get the distinguished triangle(2 . . DQ Z → DQ X → R ( j Z ′ ) ∗ DQ Z ′ +1 → . Note that Z ∩ Z is a divisor on Z , and j Z ′ : Z ′ ֒ → Z is an affine open embedding so that(2 . . p H − j R ( j Z ′ ) ∗ DQ Z ′ = 0 ( j = d ) . Since Z , Z are smooth, we then get(2 . . p H − j DQ X = R ( j Z ′ ) ∗ DQ Z ′ [ − d ] if j = d , DQ Z [ − d −
1] if j = d + 1 , . . H k ( X , p H − d DQ X ) = H k − d ( Z ′ , DQ Z ′ ) ∼ = Q if k = − d , Q if k = d − , DQ Z ′ = Q Z ′ ( d )[2 d ] with Z ′ ∼ = C d \ { } , since e Z \ Z = C d +2 \ { } which is a C ∗ -bundle over P d +1 by the natural projection, and Z ′ is its restriction over the linear subspace W ⊂ P d +1 .For X , d as in (2.1) with H ( Y ) = 0, set X := X × X , d := d + d + 1 (= dim X ) . Then(2 . . p H − j DQ X = DQ X [ − d ] ⊠ R ( j Z ′ ) ∗ DQ Z ′ [ − d ] if j = d − , DQ X [ − d ] ⊠ DQ Z [ − d −
1] if j = d , L , L ′ on X by the pull-backs of D, D ′ as is explainedin (2.1), where D ′ − D is assumed effective and base-point-free. We choose a very ampleline bundle L on X . These give very ample line bundles L , L ′ on X corresponding to the Segre embedding, since the very ample line bundles on X are obtained by the tensor product of the pull-backs of the very ample line bundles by the first and second projections from X = X × X . Condition (5) then holds by (2.1.4) and (2.2.4–6) with(2 . . j − d − , k = (2 − d ) + ( d −
1) = d − d + 1 , assuming d > d so that k >
2. By (2.2.5–6) together with the K¨unneth formula, the lastassumption implies that(2 . . H k ( d − ( X ) ∼ = H k + d − d ( X , DQ X ) if | k + d | d ,H k − d +1 − d ( X , DQ X ) if | k − d + 1 | d , So the action of c ( L ) vanishes, and the assertion is reduced to the study of the actions of c ( L ), c ( L ′ ) in (2.1). Remarks. (i) Assume d = 3, and Y, D, D ′ are as in Remark after (2.1), in particular, d = 2. Then we have j = d = 6 , k = 2 , with H ( X ) ∼ = H − ( X , DQ X ) = H ( X ) . By Corollary 1 and Proposition 3 together with (2.2.8) and (2.1.6), we see that the Lyubezniknumber λ , ( C ) is equal to 2 or 1 depending on whether the very ample line bundle is inducedfrom D or D ′ .(ii) A similar argument holds replacing Z with the inverse image of a higher codimensionallinear subspace of P d +1 . For an integer d >
2, set B := P × P d − . Let ρ ′ : Z ′ → B be the pull-back of the very ample line bundle on P d − corresponding to O P d − (1). We have the associated P -bundle ρ : Z → B. This is a compactification of Z ′ such that Z \ Z ′ is the section at infinity of ρ . Let Z be theunion of the zero-section and the section at infinity of ρ so that Z ∼ = P × P d − ⊔ P × P d − ⊂ Z. Take any smooth curve E ⊂ P of degree d E >
3. The genus g E of E is given by g E = ( d E − d E − / . (Note that the vector bundle E in (1.3) is not used in this section.) Set Z := ρ − ( E × P d − ) ⊂ Z Here dim Z = dim Z = d . Put X := Z ∪ Z ⊂ Z, Z ′ := Z \ Z , with j Z ′ : Z ′ ֒ → Z the natural inclusion. As in (2.2.2), we have the distinguished triangle(2 . . DQ Z → DQ X → R ( j Z ′ ) ∗ DQ Z ′ +1 → . Then(2 . . p H − d DQ X = DQ X [ − d ] . and we have the long exact sequence of mixed Q -Hodge structures (see [De2], [Sa2]):(2 . . → H k ( Z ) → H k ( X ) → H BM k ( Z ′ ) → H k − ( Z ) → , where H BM • denotes the Borel-Moore homology.By the Thom-Gysin sequence (1.3.2) for F • = DQ B with B := E × P d − , we have thelong exact sequence of mixed Q -Hodge structures (see Remark after (1.3)):(2 . . H k ( B ) c ′ → H k − ( B )(1) → H BM k ( Z ′ ) → H k − ( B ) c ′ → H k − ( B )(1) , where c ′ is the first Chern class of the line bundle ρ ′ : Z ′ → B . EPENDENCE OF LYUBEZNIK NUMBERS 11
Using (2.3.3–4), we can show the isomorphisms of W -graded mixed Hodge structures of odd weights(2 . .
5) Gr W odd H k ( X ) = Gr W odd H BM k ( Z ′ ) = H ( E )( d −
1) if k = 2 d − ,H ( E ) if k = 2 , W odd H k ( X ) := L i ∈ Z +1 Gr Wi H k ( X ) , etc.Indeed, the first isomorphism of (2.3.5) follows from (2.3.3). For the second, we haveGr W odd H k ( B ) = ( H ( E )( j ) if k = 2 j + 1 , j ∈ [0 , d − , W odd H BM2 d − ( Z ′ ) = Gr W odd H d − ( B )(1) = H ( E )( d − , Gr W odd H BM2 ( Z ′ ) = Gr W odd H ( B ) = H ( E ) , where the other Gr W odd H BM k ( Z ′ ) vanish. So (2.3.5) follows.Set X := X × X , d := d + d (= dim X ) , where X , d are as in (2.1). Then p H − j DQ X = ( DQ X [ − d ] ⊠ DQ X [ − d ] if j = d , d = 2 . By the last assumption in (2.1), we have H ( X ) = H ( X ) = 0 . The following morphisms are surjective (using (2.1.2)): c ( D ) , c ( D ′ ) : H ( X ) →→ H ( X ) . So there is only a difference in the rank for(2 . . c ( D ) , c ( D ′ ) : H ( X ) → H ( X ) . We have very ample line bundles L , L ′ on X defined by L := pr ∗ L ⊗ pr ∗ L , L ′ := pr ∗ L ′ ⊗ pr ∗ L . Here pr i denotes the i th projection from X × X , L is the very ample line bundle on X corresponding to D (similarly for L ′ with D replaced by D ′ , assuming D ′ − D effective andbase-point-free), and L is a very ample line bundle on X .We now show that condition (5) holds if j = j , k = k with(2 . . j := d + 1 , k := (2 − d ) + ( d −
2) = d − d (= d − , where k > d > d + 2 (= 4). The number d − H ( X ) = H − ( X , DQ X ) = H d − ( X , p H − d DQ X ) . We will show that the even weight part can be neglected effectively in condition (5) if g E ≫ W even H k ( d ) ( X ) L := L i ∈ Z Gr Wi H k ( d ) ( X ) L ( k ∈ Z ) , and similarly for Gr W odd H k ( d ) ( X ) L , Gr W even H k ( d ) ( X ) L , Gr W odd H k ( d ) ( X ) L . Here W is the weightfiltration of the canonical mixed Hodge structure on H k ( d ) ( X ) := H k ( X, p H − d DQ X ) = H d − k ( X ) . Put µ k odd ( X, L ) := µ k odd ( X ) L + µ k − ( X ) L with µ k odd ( X ) L := dim Gr W odd H k ( d ) ( X ) L , µ k − ( X ) L := dim Gr W odd H k − d ) ( X ) L , and similarly for µ k even ( X, L ), µ k even ( X ) L , µ k − ( X ) L . For L , L ′ , k as above, we then get(2 . . δ k odd := (cid:12)(cid:12) µ k odd ( X, L ) − µ k odd ( X, L ′ ) (cid:12)(cid:12) > δ k even := (cid:12)(cid:12) µ k even ( X, L ) − µ k even ( X, L ′ ) (cid:12)(cid:12) , if g E ≫
0. Indeed, δ k odd is strictly positive by (2.1.4), (2.3.5), and is proportional to g E byLemma below via the inclusion (using the K¨unneth formula):(2 . . H − d ( X , p H − d DQ X ) ⊗ H d − ( X , p H − d DQ X ) ֒ → H k ( X, p H − d DQ X ) , when the curve E is changed. On the other hand, δ k even in (2.3.8) is independent of g E . Socondition (5) holds if g E ≫
0, see also Remarks (i–iii) below for more precise arguments.This finishes the proof of Theorem 1.
Lemma.
In the above notation, δ k odd in (2 . . is proportional to g E , and δ k even remainsinvariant under the change of the plane curve E ⊂ P .Proof. The odd weight part of H • ( X ) isGr W − H ( X ) = H ( D ) , and the actions of c ( D ) , c ( D ′ ) on it vanish. We get an even weight part of H • ( X ) via theK¨unneth formula by taking the tensor product of this odd weight part with the odd weightpart of H • ( X ) which is isomorphic by (2.3.5) to(2 . . H ( E ) ⊕ H ( E )( d − . The action of c ( L ) on the latter odd weight part vanishes. Hence the actions of c ( L ) , c ( L ′ )vanish on the above tensor product, which is called the odd-odd weight part. (Here weconsider the actions on the graded pieces of the weight filtration W . Note that any morphismof mixed Hodge structures is strictly compatible with the weight filtration, and the kerneland cokernel commute with the passage to the graded quotients of the weight filtration, see[De1].) So the contribution of this odd-odd weight part vanishes by taking the differencebetween µ k even ( X, L ) and µ k even ( X, L ′ ). This shows the invariance of δ k even under the change ofthe curve E , since the even-even weight part is clearly independent of E .As for δ k odd in (2.3.8), we see that the contribution of the tensor product of the even weightpart of H • ( X ) with (2.3.10), that is, the even-odd weight part, is proportional to g E (wherethe action of c ( L ) vanish on (2.3.10)). Note that only H ( E ) in (2.3.10) contributes here,and there is no contribution of H ( E )( d −
1) for a reason of degree (since k = d − d in(2.3.7)), see also (2.3.5–6). So it remains to consider the odd-even weight part. We see thatthe odd weight part of H • ( X ) does not contribute to δ k odd in (2.3.8) by taking the tensorproduct with the even weight part of H • ( X ), since there is a difference of actions only onthe even weight part of H • ( X ) as is explained above, see also (2.3.6). This finishes the proofof Lemma. Remarks. (i) Some part of the above Lemma can be avoided if we assume, for instance, Y = P × P with D ⊂ P × P the diagonal so that H ( D ) = 0 as in Remark after (2.1). EPENDENCE OF LYUBEZNIK NUMBERS 13 (ii) Setting δ := d − >
0, we havedim H k ( Z , p H − d DQ Z ) = − | k | + δ if | k | − δ = 2 or 4 , | k | δ , k − δ ∈ Z , H k ( Z ′ , p H − d DQ Z ′ ) = g E if k = − δ − δ +2 , k = − δ − ± δ +2 ± , H k ( Z ′ , p H − d DQ Z ′ ) → H k +1 ( Z , p H − d DQ Z ) ( k = d − ± . (Note that d − ± δ +2 ± Z , X respectively replaced by Z \ Z ′ , Z , together with a morphism from thissequence to (2.3.3). (Here we also study a similar sequence for the P -bundle over P d − .)We then get H d − ( X , p H − d DQ X ) = 0 . So the contribution to δ k even in (2.3.8) via the following inclusion vanishes: H − d ( X , p H − d DQ X ) ⊗ H d − ( X , p H − d DQ X ) ֒ → H k − ( X, p H − d DQ X ) , where µ k − ( X ) L − µ k − ( X ) L ′ is involved.We have also the vanishing of the contribution to δ k even via the inclusion (2.3.9), where µ k even ( X ) L − µ k even ( X ) L ′ is involved. This follows from Remark (iii) below together with thehard Lefschetz property of the action of c ( L ) on H • ( Z ) and the commutativity of themorphisms in (2.3.3) with the action of c ( L ). (The argument is rather delicate in the case d = 4, where we need also the primitive decomposition together with (2.3.6).)For (2.3.8) it is then sufficient to assume d E > g E >
1, since dim H ( E ) = 2 g E .(iii) Let A • , B • be graded vector spaces having the actions c ′ : A i → A i +1 , c ′′ : B i → B i +1 ( i ∈ Z ), and satisfying the following conditions for some integers p, q : A i = 0 ( ∀ i / ∈ [ p, p + 2]) , c ′′ ( B i ) = B i +1 ( ∀ i ∈ [ q, q + 2]) . Set c := c ′ ⊗ id + id ⊗ c ′′ on C • := A • ⊗ B • . Then c ( C p + q +2 ) = C p + q +3 . References [BBD] Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Ast´erisque 100, Soc. Math. France,Paris, 1982.[Bo] Borel, A. et al., Algebraic D -Modules, Perspectives in Math. 2, Academic Press, Boston, 1987.[BS] Brodman, M.P. and Sharp, R.Y., Local cohomology: An algebraic introduction with geometricapplications, Cambridge Studies in Adv. Math. 60, Cambridge University Press, 1998.[De1] Deligne, P., Th´eorie de Hodge II, Publ. Math. IHES 40 (1971), 5–57.[De2] Deligne, P., Th´eorie de Hodge III, Publ. Math. IHES 44 (1974), 5–77.[Di] Dimca, A., Sheaves in topology, Universitext, Springer, Berlin, 2004.[DS] Durfee, A.H. and Saito, M., Mixed Hodge structures on the intersection cohomology of links, Compos.Math. 76 (1990), 49–67[GS] Garc´ıa L´opez, R. and Sabbah, C., Topological computation of local cohomology multiplicities, Col-lect. Math. 49 (1998), 317–324.[GM] Goresky, M. and MacPherson, R., Intersection homology II, Inv. Math. 71 (1983), 77–129.[Gr] Grothendieck, A., El´ements de g´eom´etrie alg´ebrique II, Publ. Math. IHES 8 (1961).[Ha] Hartshorne, R., Local Cohomology (A seminar given by A. Grothendieck, Harvard University, Fall,1961), Lect. Notes in Math. 41, Springer, Berlin, 1967.[Iy] Iyengar, S.B., et al., Twenty-four hours of local cohomology, Graduate Studies in Math. 87, AMSRI, 2007. [Ka] Kashiwara, M., The Riemann-Hilbert problem for holonomic systems, Publ. RIMS, Kyoto Univ. 20(1984), 319–365.[KK] Kashiwara, M. and Kawai, T., On holonomic systems of microdifferential equations III. Systemswith regular singularities, Publ. RIMS, Kyoto Univ. 17 (1981), 813–979.[KS] Kashiwara, M. and Schapira, P., Sheaves on manifolds, Springer, Berlin, 1994.[Ko] Kochmann, S., Bordism, Stable Homotopy and Adams Spectral Sequences, AMS, 1996.[Ly1] Lyubeznik, G., Finiteness properties of local cohomology modules (an application of D -modules tocommutative algebra), Inv. Math. 113 (1993), 41–55.[Ly2] Lyubeznik, G., A partial survey of local cohomology, Local cohomology and its applications (Gua-najuato, 1999), Lect. Notes in Pure and Appl. Math. 226, Dekker, New Xork, 2002, pp. 121–154.[Me] Mebkhout, Z., Une autre ´equivalence de cat´egories, Compos. Math., 51 (1984), 63–88.[NWZ] N´u˜nez-Betancourt, L., Witt, E.E. and Zhang, W., A survey on the Lyubeznik numbers, Contempo-rary Mathematics 657 (2016), 154–181.[Sa1] Saito, M., Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ. 24 (1988), 849–995.[Sa2] Saito, M., Mixed Hodge modules, Publ. RIMS, Kyoto Univ. 26 (1990), 221–333.[Swl] Switala, N., Lyubeznik numbers for nonsingular projective varieties, Bull. London Math. Soc. 47(2015), 1–6.[Swz] Switzer, R.M. Algebraic topology - homotopy and homology, Classics in Mathematics. Springer,Berlin, 2002.[Ve1] Verdier, J.-L., Dualit´e dans la cohomologie des espaces localement compacts, S´eminaire Bourbaki,Vol. 9, Exp. No. 300, Soc. Math. France, Paris, 1995, pp. 337–349.[Ve2] Verdier, J.-L., Des cat´egories d´eriv´ees des cat´egories ab´eliennes, Ast´erisque 239 (1996).[Zh] Zhang, W., Lyubeznik numbers of projective schemes, Adv. Math. 228 (2011), 575–616. T. Reichelt : Mathematisches Institut, Universit¨at Heidelberg, Im Neuenheimer Feld 205,69120 Heidelberg, Germany
E-mail address : [email protected] M. Saito : RIMS Kyoto University, Kyoto 606-8502 Japan
E-mail address : [email protected] U. Walther : Purdue University, Dept. of Mathematics, 150 N. University St., WestLafayette, IN 47907, USA
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