MONODROMY OF A FAMILY OF HYPERSURFACES CONTAINING A GIVEN SUBVARIETY ANIA OTWINOWSKA AND MORIHIKO SAITO 5 0 0 Abstract. For a subvariety of a smooth projective variety,consider the family 2 of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of n a the ambient variety and also by the cycle classes of the irreducible components J of the subvariety. Using Deligne’s semisimplicity theorem together with Steen- 6 brink’s theory for semistable degenerations, we give a simpler proof of the first author’s theorem (with a better bound of the degree of hypersurfaces) that this ] G monodromy representationis irreducible. A . h Introduction t a m ItiswellknownafterNoetherandLefschetz that,forageneralsmoothhypersurface [ X in P3, the subspace of Hodge cycles in the middle cohomology H2(X,Q) is C generated by that of P3 (i.e. it is 1-dimensional) if the degree of the hypersurface 2 C v is at least 4. This follows from the irreducibility of the monodromy representation 9 on the primitive cohomology of the hypersurfaces. Sometimes we want to consider 6 4 a family of hypersurfaces containing a given closed subvariety Z, and ask if an 4 analogue of the above assertion holds. 0 More generally, let Y be an irreducible smooth complex projective variety em- 4 0 bedded in a projective space, and Z be a (possibly reducible) closed subvariety of / h Y. Let Z{j} = {x ∈ Z : dimTxZ = j}, where TxZ denotes the Zariski tangent t space. In this paper we assume a m (0.1) dimZ +j ≤ dimY −1 for any j ≤ dimY. {j} : v In particular, 2dimZ ≤ dimY − 1. Let I be the ideal sheaf of Z, and δ be a i Z X positive integer such that I (δ)(:= I ⊗ O (δ)) is generated by global sections, Z Z Y r where O (i) denotes the restriction of O(i) on the ambient projective space. S. a Y Kleiman and A. Altman [13] then proved that condition (0.1) implies the existence of a smooth hypersurface section X of degree d of Y containing Z for any d > δ (where their condition for δ is slightly different from ours). Actually, condition (0.1) is also a necessary condition, see [19] (or (1.4) below). Let m = dimX, and Hm(X,Q)van denote the orthogonal complement of the injective image of Hm(Y,Q) in Hm(X,Q), which is called the vanishing part or the vanishing cohomology, because it is generated by the vanishing cycles of a Lefschetz pencil using the Picard-Lefschetz formula. Let Hm(X,Q)van denote the Z subspace of Hm(X,Q)van generated by thecycle classes ofthemaximal dimensional irreducible components of Z modulo the image of Hm(Y,Q) (using the orthogonal Date: Jan. 3, 2004,v.2.1. 1 2 ANIAOTWINOWSKAANDMORIHIKOSAITO decomposition) if m = 2dimZ, and Hm(X,Q)van = 0 otherwise. Let Hm(X,Q)van Z ⊥Z be its orthogonal complement in Hm(X,Q)van. The first author [19] made the following 0.2. Conjecture. Assume degX ≥ δ + 1. Then the monodromy representation on Hm(X,Q)van for the family of hypersurface sections X containing Z as above ⊥Z is irreducible. In the case Z is smooth, an easy proof was given by C. Voisin, see (1.6). In general,usingadegenerationargumentinspiredby[10],[16],thefirstauthorproved the following. 0.3. Theorem [19]. There exists a positive real number C such that Conjecture (0.2) holds if degX ≥ C(δ +1). This was used in an essential way for the proof of the main theorem in [20], which implies the Hodge conjecture for a hypersurface section of sufficiently large degree belonging to some open subset of an irreducible component of the Noether- Lefschetz locus of low codimension, and whose argument generalizes that of [18]. In the proof of Theorem (0.3), however, an asymptotic argument is used, and C can be quite large. In this paper, we give a simpler proof of the theorem below by using the theory of nearby cycles ([5], [24]) together with Hodge theory ([2], [4], [22], [25], [29]) and some cohomological properties of Lefschetz pencils ([6], [12], [14], [15], [28]). 0.4. Theorem. Let δ be as above, and d be a positive integer. Assume either d ≥ 2 + δ or a general hypersurface section of degree d − δ of Y has a nontriv- ial differential form of the highest degree. Then Conjecture (0.2) holds with the assumption replaced by degX = d. The first assumption d ≥ δ + 2 may be replaced with a weaker condition that the 2-jets at each point are generated by the global sections of O (d − δ). The Y second assumption on a differential form is stable by hypersurface sections, and it is satisfied for any d ≥ δ +1 if Y has a nontrivial differential form of the highest degree. The proofof Theorem (0.4) uses Deligne’s semisimplicity theorem [4]which implies that irreducibility is equivalent to indecomposability. We take a semistable degeneration as in [10], and calculate the graded pieces of the weight filtration on the nearby cycles using [24]. Then we can proceed by induction on m showing the nontriviality of certain extension classes by calculating the cohomology of a Lefschetz pencil or a non Lefschetz fibration. In the non Lefschetz case, we use a special kind of degeneration generalizing a construction in the surface case in [19] (see (2.7) below) if the condition d ≥ δ +2 is satisfied, and use Hodge theory (see (3.2), (3.3) below) if the second assumption on a differential form is satisfied. In the Lefschetz case, the argument is rather easy, see (2.3), (2.5) below. (We can generalizeTheorem(0.4)byreplacingO (δ),O (d−δ)withtwoamplelinebundles Y Y L , L satisfying appropriate conditions even if they are linearly independent in 1 2 the Picard group of Y tensored with Q, see (4.8) below.) By a standard argument, Theorem (0.4) implies MONODROMY OF A FAMILY OF HYPERSURFACES 3 0.5. Corollary. Under the assumption of Theorem (0.4), assume further that m = 2dimZ, and the vanishing cohomology of a general hypersurface section of degree d does not have a Hodge structure of type (m/2,m/2). Then the Hodge cycles in the middle cohomology of a general hypersurface section X of degree d of Y containing Z are generated by the image of the Hodge cycles on Y together with the cycle classes of the irreducible components of Z. In particular, the Hodge conjecture for X is reduced to that for Y. In Sect. 1, we review the theory of hypersurface sections containing a subvariety (this is mainly a reproduction from [19]). In Sect. 2, we prove the nonvanishing of some extension classes using a topological method, and in Sect. 3, we do it using a Hodge-theoretic method. In Sect. 4, we study the graded pieces of the weight filtration on the nearby cycles, and prove Theorem (0.4). In this paper, a variety means a (not necessarily reduced nor irreducible) sepa- rated scheme of finite type over C, and a point of a variety means a closed point. We say that a member of a family parametrized by the points of a variety is general if the point corresponding to the member belongs to some dense open subvariety of the variety. 1. Hypersurfaces Containing a Subvariety In this section, we review some basic facts in the theory of hypersurface sections containing a subvariety, which will be needed in the proof of Theorem (0.4). We mainly reproduce the arguments in [19], Section 1, see also [13]. Our argument works only in the case of characteristic 0. 1.1. Hypersurface sections. LetY bean(m+1)-dimensionalirreduciblesmooth projective variety embedded in a projective space P. For a positive integer i, let O (i) be the restriction to Y of O (i) on the projective space P, and define Y P Ai = Γ(Y,O (i)). Y Note that the restriction morphism Γ(P,O (i)) → Γ(Y,O (i)) P Y is not surjective in general (e.g. if Y is a hypersurface section of degree i of an abelian variety contained in P). For P ∈ Ai, we denote by X the associated P hypersurface section of degree i (in a generalized sense unless P belongs to the image of the above morphism). Let Vi = {P ∈ Ai\{0} : X is smooth}. P It is identified with a smooth variety, and Vi/C∗ parametrizes the smooth hyper- surface sections of Y of degree i. Let Z be a closed subvariety of Y, and put Ai = {P ∈ Ai : Z ⊂ X }, Vi = Ai ∩Vi. Z P Z Z In this section we do not necessarily assume condition (0.1) in Introduction. Let δ be a positive integer such that I (δ) is generated by Aδ . It is shown by Kleiman Z Z 4 ANIAOTWINOWSKAANDMORIHIKOSAITO and Altman [13] that condition (0.1) in Introduction implies the existence of a smooth hypersurface of degree j containing Z for any j > δ′, where the definition of δ′ in loc. cit. uses the ideal of Z in the projective space P instead of I (in Z particular δ ≤ δ′, and we may have a strict inequality if the degrees of the defining equations of Y in P are bigger than those for Z in Y). More precisely, they showed the following theorem for d ≥ δ′+1 and V = Adi, except possibly for the estimate i i Z of dimSingXP, which we need later. Concerning the connectedness, we will need it for the proof of Theorem (0.4) only in the cases where a ≥ 0 and either r = 1 or 2. Note that the argument is simpler in these cases because X is smooth. P1 In Theorem (1.2) below, we will take vector subspaces V of Adi, and consider i Z the following condition: (C ) d > δ and there exist a vector subspace V′ of Aδ generating O (δ) outside i i i Z Y Z and a vector subspace V′′ of Adi−δ giving an embedding of Y into a i projective space such that V is the image of V′ ⊗V′′. i i i 1.2. Theorem([13], [19]). Let a = dimY −1−max{dimZ +j} so that condition {j} (0.1) is satisfied if and only if a ≥ 0. Let d ≤ ··· ≤ d be integers with r ≤ 1 r c := codim Z. Assume d ≥ δ, and let V be a vector subspace of Adi generating Z Y i i Z IZ ⊗ OY(di). Take a general P = (P1,...,Pr) in V1×···×Vr, and put XP = T1≤i≤rXPi. Then XP is a complete intersection, dimSingXP ≤ r − a − 2, and XP \Z is smooth. Assume furthermore that dimXP ≥ 1, the above condition (Ci) is satisfied for any integer i in [c ,r] in the case where dimY ≤ 2dimZ+a+2 or Z is reduced on Z the complement of a closed subvariety of dimension < dimZ, and (C ) is satisfied i for any integer i in [cZ−1,r] otherwise. Then XP\Z is connected. Here [cZ,r] = ∅ if r < c . Z Proof. Let Π = V ×···×V , and r 1 r Σ = {(P,x) ∈ Π ×Z : rank(dP ,...,dP ) < r}, r r 1 r x where the dP are defined by taking a local trivialization of O(1), and the rank is i independent of the trivialization because the X contain Z. We have Pi T∗Z = m /(I +m2 ), x Y,x Z,x Y,x (wherem denotesthemaximalideal), andtheP ∈ V generate(I +m2 )/m2 Y,x i Z,x Y,x Y,x for any i by the definition of V (taking a local trivialization of O(1)). Let i Σ = Σ ∩(Π ×{x}). r,x r r By considering the fiber of the projection Π → Π and by induction on r we get r r−1 for any x ∈ Z , {dimY−j} codim Σ = max(j −r+1,0) ≥ j −r +1. Πr r,x By definition of a we have codim Z ≥ dimY −j +a+1, hence Y {dimY−j} codimΠr×YΣr ≥ dimY −r +a+2. MONODROMY OF A FAMILY OF HYPERSURFACES 5 For a general P ∈ Π , this implies r codim Σ ∩({P}×Z) ≥ dimY −r +a+2. Y r So it remains to show the assertion on the smoothness and the connectedness. For the smoothness, we see that a subvariety of (Y \ Z)×Π defined by the r relation x ∈ XP for (x,P) ∈ (Y \ Z)×Πr is smooth. (Indeed, in the case r = 1, the variety is defined by P t P where (P ) is a basis of V as a vector space, and i i i i 1 (t ) is the corresponding coordinate system of V . Furthermore, for any x ∈ Y \Z, i 1 some P does not vanish on a neighborhood of x, where the above equation can i be divided by this P . The argument is similar for r > 1.) Then the smoothness i follows from the Bertini theorem in characteristic 0. For the connectedness, we proceed by induction on r. Let Y′ = XP′ for a general P′ ∈ Π , where Y′ = Y if r = 1. Set b = 0 if dimY ≤ 2dimZ +a+2 or Z is r−1 reduced on the complement of a closed subvariety of dimension < dimZ, and b = 1 otherwise. Assume first r < codimYZ−b. Since Y′\XP is a smooth affine variety of dimension ≥ 2, its first cohomology with compact supports vanishes by the weak Lefschetz theorem together with Poincar´e duality. So XP is connected. By Hartshorne’s connectedness theorem (see (1.3) below) the connectedness of XP\Z is then reduced to that codimXPSingXP > 1. Since SingXP ⊂ Z, the last condition is trivially satisfied in the case r < codim Z − 1. If r = codim Z − 1 and Z is generically reduced as above so Y Y that b = 0, then XP is smooth at a general point of Z because each Vi generates I ⊗ O (d ). Thus the assertion is proved. Similarly, if r = codim Z − 1 and Z Y i Y dimY ≤ 2dimZ +a+2, then dimSingXP ≤ r−a−2 < dimZ and the assertion follows. Assume now r ≥ codim Z − b. Consider a rational map of Y′ to a projective Y space defined by the restriction of the linear system V to Y′. It induces an em- r bedding of Y′ \Z because Vr′|Y′ generates OY′(δ) outside Z and Vr′′|Y′ induces an embedding. So XP \Z is isomorphic to a general hyperplane section of Y′ \Z for this embedding. Since Y′ \Z is a smooth connected variety of dimension at least 2, the Bertini theorem in characteristic 0 implies that XP \Z is connected. This completes the proof of Theorem (1.2). 1.3. Complement to the proof of Theorem (1.2). A local complete intersec- tionisirreducibleifitisconnectedandthesingularlocushascodimension> 1. This follows from Hartshorne’s connectedness theorem (see e.g. [9], Th. 18.12), because a local complete intersection is Cohen-Macaulay. It also follows from the theory of perverse sheaves [1], considering the long exact sequence of perverse cohomology sheaves associatedto thedistinguished triangleQ → Q ⊕Q → Q →where X0 X1 X2 X3 X = X ∪ X , X = X ∩ X . Indeed, if X is a local complete intersection of 0 1 2 3 1 2 0 dimension n and X has dimension ≤ n− 2, then Q [n] is a perverse sheaf (i.e. 3 X0 pHnQ = Q [n], see e.g. [7], Th. 5.1.19.) and pHjQ = 0 for j ≥ n−1, see [1]. X0 X0 X3 So Q [n] is the direct sum of pHnQ for i = 1,2, and it is a contradiction if X X0 Xi 0 is connected. 6 ANIAOTWINOWSKAANDMORIHIKOSAITO The following gives a converse of [13] (and we reproduce the arguments here for the convenience of the reader). 1.4. Theorem [19]. Assume d ≥ δ + 1. If there exists a smooth hypersurface section of degree d containing Z, then condition (0.1) in Introduction is satisfied. Proof. Let U be a non empty smooth open subvariety of Z such that {dimY−j} dimU = dimZ and O(1) is trivialized over U. Let {dimY−j} E = Ker(T∗Y| → S T∗Z). U x∈U x It is a vector bundle of rank j over U, and the associated coherent sheaf O (E) is U generated by dP for P ∈ Aδ (where we fix a trivialization of O(1) over U). So we Z may assume that E is trivialized by P ,...,P ∈ Aδ. 1 j Z Since d ≥ δ + 1, we see that the dP for P ∈ Ad generate the 1-jets of O (E), Z U i.e. they generate O (E) ⊗ O /m2 over C for any x ∈ U (because there are U x U,x U,x smooth hypersurface sections of degree d−δ which give local coordinates of U at x). If the assumption of (1.4) is satisfied, then dP for a general P ∈ Vd gives a Z nowhere vanishing section of E over U because X is smooth. Since condition P (0.1) is equivalent to dimZ < j for any j, the assertion is reduced to the {dimY−j} following: 1.5. Lemma. Let X be a smooth variety of dimension ≥ j, and E be a trivial vector bundle of rank j over X. Let V be a finite dimensional vector subspace of Γ(X,E) which generates the 1-jets at every point of X. Then for a general σ ∈ V, the zero locus σ−1(0) ⊂ X is non empty. Proof. Since E is trivial, we may identify σ ∈ Γ(X,E) with a morphism of X to a vector space E. We have a natural morphism θ : V×X → E which sends (σ,x) to σ(x). Let Y = θ−1(0). Then it is enough to show that Y is dominant over V. Since V generates the 1-jets and dimX ≥ j, there exists y = (σ,x) ∈ Y such that the differential d σ : T X → T E = E of σ : X → E (by the above x x σ(x) identification) is surjective. Consider the commutative diagram 0 −−−→ T X −−−→ T (V×X) −−−→ T V −−−→ 0 x y σ    dxσ dyθ  y y y 0 −−−→ E −−−→ E −−−→ 0 −−−→ 0 where the top row is induced by the inclusion {σ}×X → V×X and the projection V×X → V. Then the surjectivity of d σ implies that of Kerd θ → T V by the x y σ snake lemma. So the assertion follows because Kerd θ = T Y. This completes the y y proof of Theorem (1.4). 1.6. Voisin’s proof of Conjecture (0.2) in the smooth case. Consider a rational morphism of Y to a projective space P defined by the linear system Aδ+1. Z It induces an embedding of Y \Z, in particular, it is birational to the image. Let Y be the closure of the image. If Z is smooth, we see that Y is the blow-up of Y e e along Z. For a smooth hyperplane section Y of P and the corresponding smooth es MONODROMY OF A FAMILY OF HYPERSURFACES 7 hypersurface section Y of Y, there is a morphism Hm(Y )van → Hm(Y )van by the s es s functoriality of the Gysin morphisms, and it is injective by the irreducibility of {Hm(Y )van}. Calculating the cohomology of the blow-up, we see that the dimen- es sion of its cokernel coincides with the number of the (m/2)-dimensional irreducible components of Z. So Conjecture (0.2) is proved in this case. 2. Topological Method In this section, we prove the nonvanishing of certain extension classes when the condition d ≥ δ +2 in Theorem (0.4) is satisfied. 2.1. Exact sequences. Let Y be a connected smooth complex algebraic variety, and X be a divisor on Y with the inclusion i : X → Y. Put U = Y \ X with the inclusion j : U → Y. Let f : Y → S be a proper morphism to a smooth variety S, and let g = f◦i : X → S,h = f◦j : U → S be the restrictions of f. Put m = dimY −dimS. We have a long exact sequence of constructible sheaves → Rm−1f Q γm→−1 Rm−1g Q ∗ Y ∗ X (2.1.1) → RmhQ → Rmf Q →γm Rmg Q →, ! U ∗ Y ∗ X +1 using the distinguished triangle jQ → Q → i Q → together with the functor ! U Y ∗ X Rf = Rf. Since f,g are proper, the base change holds so that the stalk of the ∗ ! direct image is isomorphic to the (relative) cohomology of the fiber. Let F = RmhQ , F′ = Cokerγ , F′′ = Kerγ , ! U m−1 m so that we have a short exact sequence of constructible sheaves (2.1.2) 0 → F′ → F → F′′ → 0. Let X = g−1(s), etc. If f,g are smooth projective and X is a hypersurface section s s of Y for s ∈ S, then we have s (2.1.3) F′ = Hm−1(X ,Q)van, F = Hm(Y ,X ,Q), F′′ = Hm(Y ,Q)prim. s s s s s s s We will assume that they are nonzero (because otherwise the extension class van- ishes). If f,g are smooth morphisms, then (2.1.1) and (2.1.2) are exact sequences of local systems, which underlie naturally variation of mixed Hodge structures, see [4], [25] (and also [22]). In the application we will also consider the dual of (2.1.2) 0 → F′′∗ → F∗ → F′∗ → 0, where ∗ denotes the dual variation of mixed Hodge structure. Note that F′∗ = F′(m−1), F′′∗ = F′′(m), F∗ = Hm(Y \X ,Q)(m), s s s where (m) denotes the Tate twist, see [4]. 2.2. Lefschetz pencils. With the above notation, assume S = P1 and g : X → S is a Lefschetz pencil of a smooth projective variety Y . Let Hm−1(X ,Q)van be 0 s the subgroup of Hm−1(X ,Q) generated by the vanishing cycles, and assume it s 8 ANIAOTWINOWSKAANDMORIHIKOSAITO nonzero. Let s be any point in the discriminant of the Lefschetz pencil. Since 0 the discriminant in the parameter space of hypersurfaces is irreducible, the last assumption is equivalent to the surjectivity of the restriction morphism (2.2.1) Hm−1(X ,Q) → Hm−1(B ,Q) s s for s sufficiently near s , where B is a small ball in X around the unique singular 0 point of X , and B := B∩X is called a Milnor fiber, see [17]. This implies that s0 s s the cospecialization morphism (2.2.2) Hm(X ,Q) → Hm(X ,Q) s0 s is an isomorphism, using a long exact sequence. So we get (2.2.3) Rjg Q is a constant sheaf on S for any j 6= m−1, ∗ X see [12], where the case j = m follows from the above argument, and the other cases are easy. As a corollary, we get in this case (2.2.4) F′′ in (2.1.2) is a constant sheaf. The above argument also implies for j = m − 1 that Rm−1g Q is a (shifted) ∗ X intersection complex, i.e. (2.2.5) Rm−1g Q = j j∗Rm−1g Q , ∗ X ∗ ∗ X where j : S′ → S is the inclusion of a dense opensubvariety over which g is smooth. (This also follows from the local invariant cycle theorem [3] or the decomposition theorem [1].) The following proposition was proved in [19] using a generalization of the Picard- Lefschetz formula together with an assertion concerning the vanishing cycles of a Lefschetz pencil and related to the classical work of Lefschetz and Poincar´e (see [14], [15], [28]). We give here a simple proof of the proposition using the above cohomological property of the Lefschetz pencil. 2.3. Proposition (Lefschetz pencil case) [19]. With the notation and the assump- tions of (2.1), assume S = P1, Y = Y ×S, f = pr , and g : X → S is a Lefschetz 0 2 pencil of Y . Let S′ be any non empty open subvariety of S over which g is smooth. 0 Then for any nonzero local subsystem G of F′′|S′, the composition of the inclusion G → F′′|S′ with the extension class defined by the restriction of the short exact sequence (2.1.2) to S′ is nontrivial as an extension of local systems. Proof. Since the local system F′′|S′ is constant, we may assume that G has rank 1, and is generated by u ∈ H0(S′,F′′|S′) = Hm(Y0,Q)prim. Assume u is the image of v ∈ H0(S′,F|S′). Then it gives a section of (2.1.2) on G. So it is enough to show that u = 0 in this case. We see that F is a (shifted) intersection complex by (2.1.2), because F′ and F′′ are (shifted) intersection complexes with support S. So H0(S,F) = H0(S′,F|S′), and we may replace S′ with S or any nonempty open subvariety of S. Thus we may assume that S′ = S \{s } and X is smooth. 0 s0 MONODROMY OF A FAMILY OF HYPERSURFACES 9 Let h′ : U′ → S′ be the restriction of h over S′, where U′ = Y′ \X′. Consider the Leray spectral sequence (2.3.1) E2p,q = Hp(S′,Rqh′!QU′) ⇒ Hp+q(Y′,X′;Q). This degenerates at E , because Ep,q = 0 unless p = 0 or 1. Thus we get w ∈ 2 2 Hm(Y′,X′;Q) whose image in H0(S′,F|S′) is v. Its image in H0(S′,F′′|S′) = Hm(Y′,Q) = Hm(Y ,Q)isu,wherethelastisomorphismfollowsfromY′ = Y ×A1. 0 0 Then the image of u in Hm(X′,Q) vanishes. But this is induced by the restriction morphism under the birationalmorphism X′ → Y . So we can verify thatu belongs 0 to the image of the Gysin morphism under the inclusion X → Y , and we get s0 0 u = 0 because u is primitive. This completes the proof of Proposition (2.3). We also give an outline of the original proof of Proposition (2.3). We start with the explanation of a generalized Picard-Lefschetz formula. 2.4. Generalized Picard-Lefschetz formula. Let F be a constructible sheaf on a curve S with a local coordinate t. Let ψ F,ϕ F denote the nearby and vanishing t t cycles, see [5]. Then we have natural morphisms can : ψ F → ϕ F, var : ϕ F → ψ F, t t t t such that (2.4.1) T −id = var◦can : ψtF → ψtF, where T is the monodromy. It is well known that the functors ψ ,ϕ commute with t t the functor assigning the dual, and duality exchanges can and var up to a sign, see e.g. [11], [21]. In the case of a Lefschetz pencil, we can identify the morphism can with the restriction to the Milnor fiber (2.2.1). Assume that ϕ F ≃ Q, and can, var are nonzero. Let γ be a generator of ϕ F, t 0 t and γ be its image in ψ F by var. Let F∗ be the (shifted) dual of F which is t defined by RHom(F,Q ). Let γ∗ be the generator of ϕ F∗ such that hγ∗,γ i = 1. S 0 t 0 0 Let γ∗ be its image in ψ F∗ by var. Then we have a generalized Picard-Lefschetz t formula (2.4.1) T(u)−u = ±hγ∗,uiγ for u ∈ ψ F, t because hγ∗,can(u)i = ±hvar(γ∗),ui, see also [6]. This was proved in [19] for the 0 0 cohomology of the complement of a hypersurface section. 2.5. Outline of the original proof of Proposition (2.3) (see [19]). It is enough to show that F|S′ has no global sections. In this case, the stalk of F∗|S′ is H (Y ,X ), and γ∗ can be constructed explicitly using the ball and the Milnor m s s fiber around the critical point (and this coincides with the construction in [14], [15], [28]), because we can identify the morphism can with the restriction to the complement of the Milnor fiber in the ball, see [19]. Furthermore, considering γ∗ at any points of the discriminant of the Lefschetz pencil, they generate H (Y ,X ). m s s (This is closely related to the classical work of Lefschetz and Poincar´e, and seems to have been known to some people, see [14]and also [15], [28].) So thelocal system F|S′ has no global section, and the assertion follows. 10 ANIAOTWINOWSKAANDMORIHIKOSAITO 2.6. Non Lefschetz fibration case. Let Y be a connected smooth projective variety embedded in a projective space P, and X be a hypersurface section of Y with at most isolated singularities. We assume m := dimX = dimY − 1 ≥ 1. Let Z be an irreducible component of X (hence Z = X if m > 1). Let d be an integer ≥ 2, and S be the parameter space of hypersurfaces of degree d of P whose intersections Z ,Y with Z,Y are smooth divisors on Z\SingX and Y respectively s s (in particular, the hypersurfaces parametrized by S do not meet SingX). Let{Hm−1(Z )van} bethelocalsubsystem of{Hm−1(Z )} generatedbythe s Z s∈S s s∈S vanishing cycles at general points of the discriminant of the morphism F Z → s∈S¯ s ¯ ¯ S, where S is the parameter space of all the hypersurfaces of degree d of P, and F Z denotes the total space of the associated family of hypersurfaces. If m > s∈S¯ s 1 and X = Z, let {Hm−1(Z )van} be the local subsystem generated by the s s∈S vanishing cycles for the inclusion Z → Y . By the Picard-Lefschetz formula, the s s latter is the orthogonal complement of the injective image of the cohomology of Y (or Y using the weak Lefschetz theorem), and hence contains the former. If s X = Z is smooth, they coincide because they are the orthogonal complement of the injective image of Hm−1(Y). If m = 1, let Hm−1(Z )van = Hm−1(Z )van(= s s Z Hm−1(Z )). e s Let S′ be a dense open subvariety of S, and L be any local system on S′ such e that {Hm−1(Zs)vZan}s∈S′ ⊂ Le ⊂ {Hm−1(Zs)van}s∈S′, and the restriction of the intersection pairing to L is nondegenerate. Let L⊥ be e e the orthogonal complement of L in {Hm−1(Z )}. Note that the restrictions of the e s intersection pairing to the injective image of Hm−1(Y) and to Hm−1(Z )van are non s degenerate using Hodge theory (or [1] because it is essentially equivalent to the hard Lefschetz theorem). Consider the kernel of the composition Hm(Y ,Z ) → Hm(Y ) → Hm+2(Y)(1), s s s where the last morphism is the Gysin morphism. Let Hm(Y ,Z )van be the quotient s s L e of the kernel by the image of L⊥. e The following is a generalization of a construction in the surface case in [19], and gives a topological proof of variants of Propositions (3.2) in the non Lefschetz case and (3.3) in the surface case. 2.7. Proposition (Non Lefschetz fibration case). With the above notation and assumptions, we have a short exact sequence of local systems on S′ (2.7.1) 0 → L → {Hm(Y ,Z )van} → {Hm(Y )van} → 0, e s s L s e and it does not split if the first and last terms are nonzero. Proof. The exactness of (2.7.1) is clear by definition. To show the non splitting of (2.7.1), we may assume S′ = S using the direct image by S′ → S. We take a smoothpointO ofX containedinZ,andconsider ahypersurface H intheambient 0 projective space which intersects X,Y transversely at smooth points outside O