The Zariski-Lefschetz principle for higher homotopy groups of nongeneric pencils

THE ZARISKI-LEFSCHETZ PRINCIPLE FOR HIGHER HOMOTOPY GROUPS OF NONGENERIC PENCILS MIHAI TIB˘ AR Abstract. We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces. 1. Introduction We find natural generators for the “vanishing homotopy” which occurs in a pencil of hypersurfaces on an analytic variety X. This result is the analogue of the Zariski-Lefschetz theorem on “vanishing cycles” for homology groups. In the same time this generalises to higher homotopy groups the Zariski-van Kampen theorem on the fundamental group. It turns out that our geometric method allows to prove such a result in a general setting, which includes for instance the use of nongeneric pencils, as we show in the following. Let Ybe a compact complex analytic space and let V⊂Ybe a closed analytic subspace such that X:= Y\Vis connected, of pure complex dimension n≥3. A pencil on Xis the restriction to X of a pencil on Y. The latter is by definition the ratio of two sections, f and g, of some holomorphic line bundle over Y. The meromorphic function h:= f/g :Y99K P1(C) is holomorphic over the complement Y\Aof the axis A:= {f=g= 0}. To define singularities of pencils in a generalised sense, we first consider a Nash blowing-up Yalong the axis (see §2), with projections p:Y→P1and σ:Y→Y. Then there is a well-defined singular locus of pwith respect to a certain natural Whitney stratification S, denoted by SingSp. We shall assume throughout this paper that dim SingSp≤0, i.e. the singularities are isolated. It should be pointed out that singularities may lie on the axis, i.e. that one may have A∩σ(SingSp)6=∅. This occurs for Date: April 13, 2005. 2000 Mathematics Subject Classification. 32S50, 14F35, 14F45, 53C40, 53C35, 58F05, 32S22. Key words and phrases. Zariski-van Kampen type theorems, nongeneric pencils, isolated singularities of functions on singular spaces, monodromy. 1 2 MIHAI TIB ˘ AR instance if the transversality of the axis to strata of Yfails at finitely many points: see Proposition 2.2 and [Ti2, §2], [Ti3, §2] for more details. Such nongeneric pencils were introduced in our preprint [Ti1] and this viewpoint allowed us to prove connectivity theorems of Lefschetz type [Ti2], and a far reaching extension of the Second Lefschetz Hyperplane Theorem [Ti3]. In the literature, examples of nongeneric pencils occured sporadically; more recently they got into light, e.g. [KPS, Ok2], precisely because one can use nongeneric pencils towards more efficient computations. For instance, M. Oka uses special pencils tangent to flex points of projective curves in order to compute the fundamental group of the complement, see e.g. [Ok1, Ok2]. Nongenericity also proved to be a useful concept for treating the topology of polynomial functions (see [NN, Ti3]) and of complements of arrangements (e.g. [LT]). A highly nongeneric situation occurs when V contains a member of the pencil, see §5.4. In order to express how the “homotopy vanishing cycles” are generated we need to introduce a homotopy variation map. Roughly, this goes as follows (see §3). Let a∈Λ = p(SingSp) be some critical value of the pencil and denote by Xa:= X∩p−1(a) the corresponding critical fibre. For a general fibre of the pencil Xcnear to Xa, we denote by X∗ athe space Xc\ ∪jBj, where Bjis a Milnor ball at some point of the set Xa∩σ(SingSp). We construct in §3 a global variation map for higher homotopy groups: (1) hvara:πq(Xc, X∗ a,¦)→πq(Xc,¦), for any q≥3. Under the condition that the involved spaces are pathconnected and that π1(Xc, X∗ a,¦) is trivial, this map commutes with the actions of π1(X∗ a,¦) and of π1(Xc,¦) via the surjection π1(X∗ a,¦)³π1(Xc,¦), respectively. In case of homology groups, one encounters variation maps in the literature ever since Picard and Lefschetz studied the monodromy around an isolated singularity of a 2-variables holomorphic function [PS, Lef]. Generalised variation maps and Picard-Lefschetz formulas play a key role in studying the topology of pencils of hypersurfaces (e.g. [Ph, Mi, La, Lo, Ga, NN], [Va1, Va2]). We have introduced in [Ti3] a global variation map in homology which generalises the local variation map to pencils with isolated singularities on arbitrarily singular (open) underlying spaces. Let us now state our general Zariski-van Kampen-Lefschetz type result which uses the homotopy variation map (1). We refer to §2 and §3 for the precise definitions of all the ingredients. The homotopy depth condition “hd” is recalled in Definition 4.5 and the notation “X∗ a” is explained in 3.4. Theorem 1.1. Let h:Y99K P1define a pencil with isolated singularities such that the axis A of the pencil is not included in Vand that the general HIGHER HOMOTOPY VARIATION 3 fibre Xcon X=Y\Vis path connected. Let us consider the following three conditions: (i) (Xc, Xc∩A)is k-connected; (ii) (Xc, X∗ a)is k-connected, for any critical value aof the pencil; (iii) hdX∩SingSpX≥k+ 2. Then we have the following two conclusions: (a) If for some k≥0the conditions (i),(ii) and (iii) hold then πq(X, Xc,¦) = 0 for all q≤k+ 1. (b) If for some k≥2the conditions (i) and (ii) hold, and if condition (iii) is replaced by the following one: (iii’) hdX∩SingSpX≥k+ 3, then the kernel of the surjective map πk+1(Xc,¦)³πk+1(X, ¦)is generated by the images of the variation maps hvara. The conditions in our theorem are satisfied by a large class of spaces X. For instance, the conditions (ii) and (iii), resp. (iii’), are both fulfilled as soon as Xis a singular space which is a locally complete intersection of dimension n≥k+2, resp. n≥k+3 (see §5.1). We send to §5 for many other comments on these conditions and on what they become in more particular situations. In what concerns Part (a) of Theorem 1.1, this is a general “first Lefschetz hyperplane theorem” for homotopy groups. It has been proved in some particular cases in the past and it recovers, with significantly weakened hypotheses, our recent result [Ti2]. The focus of this paper is actually on Part (b) of Theorem 1.1, which treats the “homotopy vanishing cycles”. Our approach is in the spirit of the Lefschetz method [Lef], as presented by Andreotti and Frankel in [AF]. Since the homology conterpart of our result (which is easier) is presented in [Ti3], we concentrate here on the more delicate constructions and ingredients of the proof which are needed in case of homotopy groups. We use homotopy excision (the Blackers-Massey theorem), more precisely the general version proved by Gray [Gr, Cor; 16.27], and the comparison between homotopy and homology groups via the Hurewicz map. Our result does not cover the case k= 1, hence the homotopy group π2, because of the technical restriction in the definition of the variation map (1). The cornerstone of the proof is perhaps Proposition 4.9 in which we identify the homotopy variation map to the boundary map of the long exact sequence of the pair (X∗ D, Xc), by expanding Milnor’s explanation of the Wang exact sequence [Mi, pag. 67]. All those techniques are coupled with the general theory of stratified singular spaces and of Milnor fibrations of stratifies singularities of holomorphic functions (see e.g. [Lˆe, GM]). Notably 4 MIHAI TIB ˘ AR Proposition 4.8 is a key interplay between the use of the singular Milnor fibration and the homotopy excision. Theorem 1.1 represents a synthetic viewpoint on Zariski-van Kampen type results for higher homotopy groups, having in the background the pioneering work of Lefschetz [Lef], Zariski and van Kampen [vK]. It recovers several cases considered before in the literature and in the same time it extends the range of applicability of the Zariski-van Kampen-Lefschetz principle. For instance we show in §5.3 how to get a version of Theorem 1.1 in the case of complements of singular projective spaces, which seems to be a new result. Some other possible applications of Theorem 1.1 parallel the ones for homology groups which have been discussed in [Ti2, Applications] and [LT]. Such a progress would not have been possible without the contribution of a long list of articles around the Lefschetz slicing principle, from which we took a part of the inspiration, such as [AF, Ch1, Ch2, Fu, FL, GM, HL1, HL2, HL3, La, Lef, Li, Lo, Mi]. Results of Zariski-van Kampen type have been proved by Libgober for generic Lefschetz pencils of hyperplanes and in the particular case when Xis the complement in Pnor Cnof a hypersurface with isolated singularities [Li]. D. Ch´eniot, A. Libgober and C. Eyral authored in 2002 two preprints [CL, CE] treating, with different background, interesting particular cases of the problem we are concerned with here. This paper completes in detail a certain part of the program “Lefschetz principle for nongeneric pencils” presented in our 2001 preprint [Ti1]. We own warm thanks to several institutions from which we have got hospitality in connection to this project: Newton Institute (2000), IAS Princeton (2002), CRM Barcelona (2004). 2. Pencils with singularities in the axis As denoted before, let X:= Y\Vbe the difference of two compact complex analytic spaces. We introduce the basic notations and recall from [Ti2] what are the singularities of a pencil, in a stratified sense. Consider the compactification of the graph of h, namely the space: Y:= closure{y∈Y, [s;t]∈P1|sf(y)−tg(y) = 0} ⊂ Y×P1. Let us denote X:= Y∩(X×P1). Consider the projection p:Y→P1, its restriction p|X:X→P1and the projection to the first factor σ:Y→Y. Notice that the restriction of pto Y\(A×P1) can be identified with h. Let Wbe an analytic Whitney stratification of Ysuch that Vis union of strata. Its restriction to the open set Y\Ainduces a Whitney stratification HIGHER HOMOTOPY VARIATION 5 on Y\(A×P1), via the mentioned identification. We denote by Sthe coarsest analytic Whitney stratification on Ywhich coincides over Y\(A×P1) with the one induced by Won Y\A. This stratification exists within a neighbourhood of A×P1, by usual arguments (see e.g. [GLPW]), hence such stratification is well defined on Y. We call it the canonical stratification of Ygenerated by the stratification Wof Y. Definition 2.1. [Ti2] One calls singular locus of pwith respect to Sthe following subset of Y: SingSp:= [ Sβ∈S Singp|Sβ. The set Λ := p(SingSp) is called the set of critical values of pwith respect to S. We say that the pencil defined by h:= f/g :Y99K P1is a pencil with isolated singularities if dim SingSp≤0. Remark that SingSpis closed analytic. Remark also that pis proper and analytic in a neighbourhood of A×P1and that Shas finitely many strata. Then, by a Bertini-Sard result which is a consequence of the First Isotopy Lemma [Th], see also [Ve], it follows that Λ ⊂P1is a finite set and that the maps p:Y\p−1(Λ) →P1\Λ and p|X:X\p−1(Λ) →P1\Λ are stratified locally trivial fibrations. In particular, h:Y\(A∪h−1(Λ)) →P1\Λ is a locally trivial fibration. Such nongeneric pencils may occur whenever the axis Ais not in general position at finitely many points. More precisely, we have the following example of situation: Proposition 2.2. [Ti2, Prop. 2.4] Let X=Y\V⊂PNbe a quasi-projective (singular) variety and let ˆ h=ˆ f/ˆg define a pencil of hypersurfaces in PNwith axis ˆ A. Let Cdenote the set of points on ˆ A∩Ywhere some member of the pencil is singular or where ˆ A is not stratified transversal to Y. If dim C≤0and if the stratified critical points of the restriction ˆ h|:Y\A→P1are isolated, then dim SingSp≤0. ¤ We shall assume throughout the paper that our pencil on Xhas isolated singularities. Then, by the compactness of Y, the set SingSpis a finite set. Let us emphasize that the points of SingSpare not necessarily contained in Xand that some of them may be on the blown-up axis A×P1. 3. Homotopy variation maps Inspired by the homology constructions throughout the literature and by a particular Zariski-van-Kampen type theorem for higher homotopy groups 6 MIHAI TIB ˘ AR proved in the beginning of the 1990’s by Ligbober [Li], we have given in the preprint [Ti1] (unpublished) the main lines of the construction of global homotopy variation maps for pencils with isolated singularities. We complete here in full detail this construction1. For any M⊂P1, we denote YM:= p−1(M) and XM:= X∩YMand XM:= σ(XM). Let ai∈Λ and let SingSp={aij }i,j ⊂Y, where aij ∈Yai. For c∈P1\Λ we say that Yc, respectively Xc, respectively Xc, is a general fibre. 3.1. Milnor fibration at isolated stratified critical points of holomorphic functions. At some singularity aij, in local coordinates, we take a small ball Bij centered at aij. For small enough radius of Bij, this is a Milnor ball2of the local holomorphic germ of the function pat aij . Next we may take a small enough disc Di⊂P1at ai∈P1, so that (Bij, Di) is Milnor data for pat aij . Moreover, we may do this for all (finitely many) singularities in the fibre Yai, keeping the same disc Di, provided that it is small enough. Since the function p:Y→P1has isolated stratified singularities, the fibres of pare endowed with the stratification induced by Sexcept that we have to introduce the point-strata {aij}. Every fibre has therefore a natural induced Whitney stratification. By the general theory of holomorphic function germs [GM, Lˆe] we have that the restriction p|:Y∂¯ Di→∂¯ Diis a locally trivial stratified fibration over the circle ∂¯ Di. Since p:Y→P1has a stratified isolated singularity at aij, the sphere ∂¯ Bij is stratified transversal to all the fibres of pabove the points of Di. It follows that the restriction of pto the pair of spaces (YDi\ ∪jBij,YDi∩ ∪j∂¯ Bij) is a trivial stratified fibration over Di. 3.2. Geometric monodromy. Let’s choose ci∈∂¯ Di. One constructs in the usual way (see Looijenga’s similar discussion in [Lo, 2.C, page 31]), by using a stratified vector field, a characteristic morphism hi:Xci→ Xciof the fibration over the circle p|:X∂¯ Di→∂¯ Di, which is a stratified homeomorphism and it is the identity on Xci\ ∪jBij. This is called the geometric monodromy of the locally trivial fibration p|:X∂¯ Di→∂¯ Diand corresponds to one counterclockwise loop around the circle ∂¯ Di. Clearly hi is not uniquely defined with these properties, but its relative isotopy class in the group Iso(Xci, Xci\ ∪jBij) of relative isotopy classes of stratified homeomorphisms which are the identity on Xci\ ∪jBij is unique. 1A similar construction in case of homology groups appears in [Ti3]. 2One may consult [GM, Lˆe] for the definition and the terminology concerning the local Milnor fibration in case of a holomorphic function on a singular analytic space, at an isolated critical point with respect to a given stratification of the space. HIGHER HOMOTOPY VARIATION 7 3.3. Construction of the homotopy variation map. We assume in the rest of this section that Xci\ ∪jBij is path connected, for all i. This assumption is actually fulfilled under the hypotheses of our Theorem 1.1(b). More precisely, the fact that the general fibre Xcis path connected and that (Xci, Xci\ ∪jBij) is 1-connected imply that Xci\ ∪jBij is path connected. Let γ: (Dq, Sq−1, u)→(Xci, Xci\ ∪jBij, v), for q≥3, be some continuous map, and let [γ] be its homotopy class in πq(Xci, Xci\ ∪jBij, v). Note that [hi◦γ] is a well-defined element of πq(Xci, Xci\ ∪jBij, v) and does not depend on the representative hiin its relative isotopy class. Consider the map ρ: (Dq, Sq−1, u)→(Dq, Sq−1, u) which is the reflection into some fixed generic hyperplane through the origin, which contains u. Then [γ◦ρ] is the inverse −[γ] of [γ]. We use the additive notation since the relative homotopy groups πqare abelian for q≥3. Consider now the map: (2) µi:πq(Xci, Xci\ ∪jBij, v)→πq(Xci, Xci\ ∪jBij, v) defined as follows: µi([γ]) = [(hi◦γ)∗(γ◦ρ)] = [hi◦γ]−[γ]. The notation “∗” stands for the operation on maps which induces the group structure of the relative πq. The map µiis well-defined and it is an automorphism of πq(Xci, Xci\ ∪jBij, v), because of the abelianity of the relative homotopy groups πq, for q≥3. For the time being, we cannot extend this construction to q= 2. In case q= 1 there are the monodromy relations which enter in the well-known Zariski-van Kampen theorem. Claim:the map νi:= (hi◦γ)∗(γ◦ρ): (Dq, Sq−1, u)→(Xci, Xci\∪jBij, v) is homotopic, relative to Sq−1, to a map (Sq, u)→(Xci, v). To prove the claim, we first observe that for the restriction of γto Sq−1, which we shall call γ0, we have hi◦γ0=γ0, since the geometric monodromy hiis the identity on im γ0. Without loss of generality, we may and shall suppose in the following that the geometric monodromy hiis the identity on a small tubular neighbourhood Tof Xci\ ∪jBij within Xci. Let ν0 idenote the restriction of νito Sq−1. We claim that one can shrink im ν0 ito the point vthrough a homotopy within im ν0 i. The precise reason is that, by the definition of ρ, the map γ0∗(γ0◦ρ) is homotopic to the constant map to the base-point v, through a deformation of the image of γ0∗(γ0◦ρ) within itself. We may moreover extend this homotopy to a continuous deformation of the map νiin a small collar neighbourhood τ of Sq−1in Dq, which has its image in the tubular neighbourhood T, such that it is the identity on the interior boundary of τ(which is a smaller sphere Sq−1 1−ε⊂Dq). Then we extend this deformation as the identity to the rest of Dq. In this way we have constructed a homotopy between νi and a map (Dq, Sq−1, u)→(Xci, v, v), which is nothing else than a map 8 MIHAI TIB ˘ AR (Dq, u)→(Xci, v). This represents an element of the group πq(Xci, v), so our claim is proved. This construction is well-defined up to homotopy equivalences, hence it yields a map: (3) hvari:πq(Xci, Xci\ ∪jBij, v)→πq(Xci, v), which we call the (homotopy) variation map of hi. Let us point out that hvariis a morphism of groups. Indeed, due to the abelianity we have: hvari([γ] + [δ]) = [(hi◦γ)∗(hi◦δ)] −[γ∗δ] = [hi◦γ] + [hi◦δ]−[γ]−[δ] = [hi◦γ]−[γ]+[hi◦δ]−[δ] = hvari([γ]) + hvari([δ]). By its geometric definition, this variation map enters in the following commuting diagram: (4) πq(Xci, Xci\ ∪jBij, v)µi −→ πq(Xci, Xci\ ∪jBij, v) hvari& % j] πq(Xci, v) where j: (Xci, v)→(Xci, Xci\ ∪jBij, v) is the inclusion. 3.4. Convention of notations. In local coordinates at aij,Yaiis a germ of a stratified complex analytic space; hence, for a small enough ball Bij, the set ¯ Bij ∩Xai\ ∪jaij retracts to ∂¯ Bij ∩Xai, by the local conical structure of stratified analytic sets [BV]. It follows that Xai\ ∪jaij is homotopy equivalent, by retraction, to Xai\ ∪jBij. We have seen in §3.1 that the restriction p|:XDi\ ∪jBij →Diis a trivial stratified fibration. This implies that the restriction to X, i.e. p|:XDi\ ∪jBij →Diis a trivial stratified fibration too. Since Xai\ ∪jBij and Xci\ ∪jBij are both fibres of this fibration, we shall identify them via a fixed trivialisation along a chosen path in Di. In the remainder of this paper, for the sake of simplicity, we shall denote the spaces Xci\ ∪jBij and Xai\ ∪jBij by the same symbol X∗ ai. In particular, Xai\ ∪jBij becomes in this way a subspace of Xci. We shall use in the remainder the following notation for the variation map, only as a symbolic replacement for (3), but not having an intrinsic meaning: (5) hvari:πq(Xci, X∗ ai,¦)→πq(Xci,¦), for any q≥3. This is actually the notation used in the statement of Theorem 1.1. HIGHER HOMOTOPY VARIATION 9 3.5. The action of π1.We have the standard action of π1(Xc,¦) on the higher homotopy groups πq(Xc,¦) and on any relative homotopy group where Xcappears at the second position. Moreover, the hypotheses of Theorem 1.1(b) imply that X∗ aiis path connected for all i, as explained in §3.3, and that π1(X∗ ai,¦)i∗ →π1(Xci,¦) is an isomorphism. We shall therefore identify the two actions modulo the isomorphism i∗. So one has the action of π1(X∗ ai,¦) on those higher relative homotopy groups of pairs where X∗ aiis the second space of the pair, and one also has the action of π1(X∗ ai,¦), via the isomorphism i∗, on πq(Xci,¦) and on the higher relative homotopy groups of pairs where Xciis the second space of the pair. Let’s then prove that the image of hvariis invariant under the action of π1(Xci,¦). For some β∈π1(X∗ ai,¦), we have hvari(β[γ]) = β[hi◦γ]−β[γ] = βhvari([γ]) since, by the functoriality, the action of βcommutes with the morphism induced by the geometric monodromy map hi:Xci→Xci. We shall denote in the following by ˆπqthe quotient of the corresponding higher homotopy group by the action of π1(Xc,¦) or of π1(X∗ ai,¦), eventually through the isomorphism i∗. We shall use the same notation ˆνfor the passage to the quotients of some morphism of groups νwhich commutes with the action of π1. 4. Proof of Theorem 1.1 We shall prove in parallel (a) and (b). The only proof of (a) would be much simpler and can be easily extracted. Relative to the claim (b), let us first remark that the kernel of πk+1(Xc,¦)³πk+1(X, ¦) is invariant under the the π1(Xc,¦)-action on πk+1(Xc,¦), by the functoriality of this action. So in order to prove (b) it suffices to show that the image of ˆ ∂: ˆπk+2(X, Xc,¦)→ˆπk+1(Xc,¦) is generated by the images im( [ hvari), since im(hvari) is also invariant under the π1-action, see §3.5. This is what we shall do in the last part of the proof below. When dealing with homotopy groups, we shall need to apply the homotopy excision theorem of Blackers and Massey in the form given by Gray [Gr, Cor; 16.27]. For the connectivity claim (a), we follow in the beginning the lines of [Ti2]. Let A0:= A∩Xcand assume that A06=∅. Let K⊂P1be a closed disc with K∩Λ = ∅and let Ddenote the closure of its complement in P1. We denote by S:= K∩ D the common boundary, which is a circle, and take a point c∈S. Then take standard paths ψi(non self-intersecting, non mutually intersecting) from cto ci, such that ψi⊂ D \ ∪iDi. The 16 MIHAI TIB ˘ AR 5. Some special cases of the main theorem The Lefschetz method for generic pencils of hyperplanes and the comparison with the axis of the pencil have been used before by several authors in order to prove connectivity results of increasing generality. Our Theorem 1.1(a) improves [Ti2] by weakening the hypotheses. As we have explained in loc.cit., our result [Ti2] recovered in its turn other previous connectivity results [La, Ch1, Ch2, Ey]. In the following we show what becomes Theorem 1.1(b) in some particular cases. 5.1. Aspects of the homotopy depth condition. The condition (iii) of Theorem 1.1 is satisfied as soon as Xis a locally complete intersection of dimension ≥k+ 2. This follows from the fact that the rectified homotopical depth4is ≥k+2 for such an X(cf [HL3]). Indeed, one can easily show that, more generally, the condition rhd X≥k+ 2 implies the condition (iii), i.e. that hdX∩SingSpX≥k+ 2.5Indeed, rhd X≥k+ 2 implies rhd X≥k+ 2, since we may apply [HL3, Theorem 3.2.1] to the hypersurface Xof X×P1. This in turn implies hdvX≥k+ 2, for any point v∈X, by definition. Moreover, rhd X≥k+ 2 implies that the pair (XDi, Xci) is (k+ 1)- connected, by [Ti2, Proposition 4.1]. This provides a shortcut to the proof of Part (a) of Theorem 1.1, in the sense that the condition (ii) is not needed, and therefore Lemma 4.6 and Proposition 4.8 are useless. In what concerns Part (b) of Theorem 1.1, the condition rhd X≥k+ 3 implies not only (iii’), as shown above, but also the condition (ii), which is important in the proof of Part (b). Indeed, the k-connectivity of the pair (Xci, X∗ ai) follows from a sequence of homotopy excisions, which reduces the problem to proving the k-connectivity of each pair (Xci∩Bij , Xci∩∂Bij ). This is true since Xci∩Bij is (k+1)-connected and Xci∩∂Bij is k-connected. The former follows from [Ti2, Proposition 3.4] which is based on [HL3, Corollary 4.2.2]. The proof goes as follows: rhd X≥k+3 implies rhd Xai≥ k+2, by [HL3, Theorem 3.2.1]. This implies hdaij Xai≥k+2, which shows that Xai∩∂Bij is k-connected. Finally, we already know that Xai∩∂Bij is homeomorphic to Xci∩∂Bij, so this ends the proof of our claim. 5.2. Pencils on Xhaving no singularities in the axis. Let us consider the situation (A×P1)∩X∩SingSp=∅. We say that “Xdoes not contain singularities along the axis of the pencil”. Even so, there might still be singularities in the axis on Y, and they certainly influence the topology of 4see [Ti2] and the source [HL3] for the definition and properties of this notion which goes back to Grothendieck; in particular, rhd does not depend on the stratification. 5see also [Ti3] for a similar remark in homology. HIGHER HOMOTOPY VARIATION 17 the pencil on X. However, this situation be further investigated as follows, see also §5.4 for an interesting case. We claim that the homotopy depth condition (iii), resp. (iii’), may be replaced by the following more general, global condition: (15) (X, X \Σ) is (k+ 2)-connected, resp. (k+ 3)-connected, where Σ := σ(SingSp)⊂Y. Then our Theorem 1.1(a) specializes to a connectivity statement which recovers Eyral’s main result [Ey], where such a condition was used for generic pencils. Indeed, we notice that the homotopy depth condition was used only when comparing XDito X∗ Di. We may therefore get rid of this comparison by replacing everywhere in the proof of Theorem 1.1 the space Xby the space X\Σ. In particular, this reduces to tautologies Lemma 4.6 and Corollary 4.7. We then get the conclusion of Theorem 1.1 for the inclusion Xc,→X\Σ instead of the inclusion Xc,→X. But, at this stage, we may just plug in Xin the place of X\Σ since the condition (15) tells that X\Σ and Xhave isomorphic homotopy groups up to πk+1, and this is all we need. Still in case (A×P1)∩X∩SingSp=∅, let us observe that if the condition (i) of Theorem 1.1 is fulfilled, then the condition (ii) becomes equivalent to: (16) (X∗ ai, A ∩X∗ ai) is (k−1)-connected. Indeed, when there are no singularities in the axis, we have A∩X∗ ai= A∩Xc, for any iand our claim follows from the exact sequence of the triple (Xc, X∗ ai, A ∩X∗ ai). We have to point out that even if there are “no singularities 5.3. The classical case: generic pencils on the projective space. From the preceding observations on pencils without singularities in the axis, one may derive the following statement for complements of arbitrarily singular subspaces in Pn. Its proof goes by induction, based on the abundance of generic pencils, i.e. pencils with no singularities along the axis and such that the axis Ais transversal to all strata and is not included into the subspace V. The proof actually follows the pattern of the homology proof discussed in [Ti3, §4.3], supplemented by our homotopy considerations in §4 above. Corollary 5.1. Let Y=Pn. Let V⊂ Y be a singular complex algebraic subspace, not necessarily irreducible. For any hyperplane H⊂ Y transversal to all strata of V, we have: (a) H∩(Y \ V),→ Y \ Vis a (n+ codimV−2)-equivalence. (b) There exist generic pencils of hyperplanes having Has a generic member. If dim V≤2n−5then the kernel of the surjection 18 MIHAI TIB ˘ AR πn+codimV−2(H∩(Y \V),¦)³πn+codimV−2(Y \ V, ¦)is generated by the images of the variation maps hvariof such a generic pencil. ¤ Proof. The part (a) of this Corollary is well-known, see e.g. [Ch1, Ey], and its proof goes by induction on dim V, since by repeatedly slicing we arrive to the case dim V= 0. The conditions of Theorem 1.1 are satisfied at each induction step, as follows. The generic slice Xcis path connected at each step. Condition (iii), resp. (iii’), is empty at every step, since a generic pencil has no singularities in Pn−j\V. Condition (ii) is implied by (i) since it is equivalent to (16), as shown above. The condition (i) in case dim V= 0, is the level of connectivity of the pair (Pn−dim V−1,Pn−dim V−2), which is known to be equal to k= 2(n−dim V−1) −1. Then Theorem 1.1(a) gives the level of connectivity k+ 1 of the higher pair (Pn−dim V\ V∩Pn−dim V,Pn−dim V−1), which becomes the condition (i) for the next induction step. This proves part (a). Part (b) also follows from Theorem 1.1(b), since we have seen that the conditions are inductively fulfilled. In addition, we have to insure that the initial step verifies the requirement k≥2. This amounts to dim V≤2n−5. ¤ The part (b) of this Corollary seems to be a new result since we consider the general situation of complements of arbitrarily singular subspaces. There is a single case which is not covered by the condition dim V≤2n−5, namely the case n= 3, dim V= 2 (since hvar is not well defined for π2). Of course, the case n= 2 does not matter here since it is covered by the classical Zariski-van Kampen theorem. In the particular case V⊂Cnis a hypersurface with at most isolated singularities and transversal to the hyperplane at infinity, a similar type of result, but with different background, has been proved by Libgober [Li], including at the π2level. 5.4. A complementary case. It appears that the case when Vcontains a member of the pencil, which was excluded in our main theorem (since A⊂Vin this case), can also be treated. For homology groups, this was explained in [Ti3, §4-5]. Our main result can be reformulated, with much less restrictive conditions. Theorem 5.2. Let h:Y99K P1define a pencil with isolated singularities, such that Vcontains a member of the pencil. Le Xcbe path connected and let (Xc, X∗ ai)be k-connected, for a general member Xcand any atypical one X∗ ai. Then: (a) If k≥0and (X, X \Σ) is (k+ 1)-connected, then the inclusion Xc,→Xis a (k+ 1)-equivalence. 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