Hypersurface Singularities and the Swing

Hypersurface Singularities and the Swing∗ arXiv:math/0411432v4 [math.AG] 7 Feb 2005 Lê Dũng Tráng and David B. Massey Abstract Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of f . This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves. 1 Introduction and Previous Results Let U be an open neighborhood of the origin in Cn+1 , and let f : (U, 0) → (C, 0) be complex analytic. We shall always suppose that dim0 Σf = 1, unless we explicitly state otherwise. Let Ff = Ff,0 denote the Milnor fiber of f at the origin. It is well-known (see [6]) that the reduced e ∗ (Ff ), of Ff can be non-zero only in degrees n − 1 and n, and is free Abelian in degree integral homology, H e n−1 (Ff ) and H e n (Ff ); in n. For arbitrary f , it is not known how to calculate, algebraically, the groups H fact, it is not known how to calculate the ranks of these groups. However, there are a number of general results known for these “top” two homology groups of Ff . First, we need to make some choices and establish some notation. We assume that the first coordinate z0 on U is a generic linear form; in the terminology of [11], we need for z0 to be “prepolar” (with respect to f at the origin). This implies that, at the origin, f0 := f|V (z0 ) has an isolated critical point, that the polar curve, Γ := Γ1f,z0 , is purely 1-dimensional at the origin (which vacuously includes the case Γ = ∅), and Γ has no components contained in V (f ) (this last property is immediate in some definitions of the relative polar curve). For convenience, we assume throughout the remainder of this paper that the neighborhood U is re-chosen, if necessary, so small that Σf ⊆ V (f ), and every component of Σf and Γ contains the origin. Now, there is the attaching result of Lê from [9] (see, also, [11]), which is valid regardless of the dimension of the critical locus: ◦ Theorem 1.1. Up to diffeomorphism, Ff is obtained from D ×Ff0 by attaching τ := Γ · V (f ))0 handles of index n. ∗ The second author would like to thank the Abdus Salam ICTP for their hospitality; most of this paper was written during a visit there. AMS subject classifications 32B15, 32C35, 32C18, 32B10. keywords: hypersurface singularity, Milnor fiber, swing, polar curve, vanishing cycles, discriminant, Cerf diagram, intersection diagram, perverse sheaves 1 Remark 1.2. On the level of homology, Lê’s attaching result is a type of Lefschetz hyperplane result; it says e i (Ff ), e i (Ff0 ) ∼ that, for all i < n − 1, the inclusion map Ff0 = Ff ∩ V (z0 ) ֒→ Ff induces isomorphisms H =H e n (Ff ) and H e n−1 (Ff ) are, respectively, isomorphic to the kernel and cokernel of the boundary map and H ∂ e n−1 (Ff0 ) ∼ Zτ ∼ = Zµf0 , = Hn (Ff , Ff0 ) −→ H where µf0 denotes the Milnor number of f0 at the origin. Therefore, one can certainly calculate the difference of the reduced Betti numbers of Ff : b̃n (Ff ) − b̃n−1 (Ff ) = τ − µf0 . Hence, bounds on one of b̃n (Ff ) and b̃n−1 (Ff ) automatically produce bounds on the other. We remind the reader here of the well-known result, first proved by Teissier in [15] (in the case of an isolated singularity, but the proof works in general), that    ∂f  τ = Γ · V (f ))0 = Γ · V + Γ · V (z0 ))0 . ∂z0 0 As defined in [11], the first summand on the right above is λ0 := λ0f,z0 (0), the 0-dimensional Lê number, and 1 second summand on the right above is γ 1 := γf,z (0), the 1-dimensional polar number. 0 ◦ For each component ν of Σf , let µν denote the Milnor number of f|V (z0 −a) at a point close to the origin on ν ∩ V (z0 − a), where a is a small non-zero complex number. Then, λ1 := λ1f,z0 (0) := X◦ µν ν · V (z0 ))0 ν is the 1-dimensional Lê number of f . Now, it is well-known, and easy to show that µf0 = γ 1 + λ1 . Again, see [11] for the above definitions and results. In Proposition 3.1 of [11], the second author showed how the technique of “tilting in the Cerf diagram” or “the swing”, as used by Lê and Perron in [10] could help refine the result of Theorem 1.1. Here, we state only the homological implication of Proposition 3.1 of [11]. ∂ e n−1 (Ff0 ) maps a direct summand of Hn (Ff , Ff0 ) of Theorem 1.3. The boundary map Hn (Ff , Ff0 ) −→ H 1 e rank γ isomorphically onto a direct summand of Hn−1 (Ff0 ). e n (Ff ) is at most λ0 , and the rank of H e n−1 (Ff ) is at most λ1 . Thus, the rank of H However, if one of the components ν of Σf is itself singular, then the above bounds on the ranks are known not to be optimal. A result of Siersma in [14], or an easy exercise using perverse sheaves (see the remark at the end of [14]), yields: ◦ e n−1 (Ff ) is at most P µν . Theorem 1.4. The rank of H ν 2 P ◦ Of course, if all of the components ν of Σf are smooth, and z0 is generic, then λ1 = ν µν , and the bounds on the ranks obtained from Theorem 1.3 and Theorem 1.4 are the same. In addition, Theorem 1.4 e n−1 (Ff ). We should is true with arbitrary field coefficients; this yields bounds on the possible torsion in H also remark that the result of Siersma from [14] that we cite above can actually yield a much stronger bound if one knows certain extra topological data – specifically, one needs that the “vertical monodromies” are non-trivial. e n−1 (Ff ) = λ1 ? Now, in light of Theorem 1.3 and Theorem 1.4, the question is: Is it possible that rank H Of course, the answer to this question is “yes”; if f has a smooth critical locus which defines a family of 1 e n (Ff ) = 0 and H e n−1 (Ff ) ∼ isolated singularities with constant Milnor number µf0 , then certainly H = Zλ = Zµf0 . We refer to this case as the trivial case. It is important to note that being in the trivial case implies that V (z0 ) transversely intersects the smooth critical locus at the origin. By the non-splitting result, proved independently by Gabrielov [5], Lazzeri [7], and Lê [8], we have: Proposition 1.5. The trivial case is equivalent to the case Γ = ∅. Now, we can state our Main Theorem: Main Theorem. Suppose that dim0 Σf = 1 and dim0 Σf0 = 0. Then, the following are equivalent: a) We are in the trivial case, i.e., f has a smooth critical locus which defines a family of isolated singularities with constant Milnor number µf0 ; e n−1 (Ff ) = λ1 ; b) rank H e n−1 (Ff ; Z/pZ) = λ1 . c) there exists a prime p such that dim H e n−1 (Ff ) < λ1 , and so rank H e n (Ff ) < λ0 , and these Thus, if we are not in the trivial case, rank H inequalities hold with Z/pZ coefficients (here, p is prime). Remark 1.6. We remark again that if one of the components of Σf is itself singular (and, hence, we are not e n−1 (Ff ) < λ1 already follows from Theorem 1.4. Even in the trivial case), then the conclusion that rank H e n−1 (Ff ) < λ1 from in the case where all of the components of Σf are smooth, we could conclude that rank H [14] if we knew that one of the vertical monodromies were non-trivial. However, the vertical monodromies are fairly complicated topological data to calculate, and it is also true that the vertical monodromies can be trivial even when the polar curve is non-empty, i.e., when we are not in the trivial case. Thus, our Main Theorem cannot be proved by analyzing the vertical monodromies. In [13], Siersma proved another closely related result. On the level of homology, what he proved was ◦ that, if we are not in the trivial case, and Σf has a single smooth component, ν, such that µν = 1, then e n−1 (Ff ) = 0; our Main Theorem, including the modulo p statement, is a strict generalization of this. H In addition, we should point out that, in [3], Th. de Jong provides evidence that a result like our Main Theorem might be true. 3 We prove our Main Theorem by combining the swing technique of Theorem 1.3 and the connectivity of the vanishing cycle intersection diagram for isolated singularities, as was proved independently by Gabrielov in [5] and Lazzeri in [7]. In some recent notes, M. Tibăr uses similar techniques and reaches a number of conclusions closely related to our result. As a corollary to our Main Theorem, we show that it implies that the vanishing cycles of f , as an object in the category of perverse sheaves, cannot be semi-simple in non-trivial cases where Σf has smooth components of arbitrary dimension. In the final section of this paper, we make some final remarks and present counterexamples to some conceivable “improvements” on the statement of the Main Theorem. 2 The Swing In the Introduction, we referred to the swing (or, tilting in the Cerf diagram), which was used by Lê and Perron in [10] and in Proposition 3.1 of [11], where the swing was used to prove Theorem 1.3. The swing has also been studied in [2], [16], [11], [17]. As the swing is so crucial to the proof of the main theorem, we wish to give a careful explanation of its construction. Suppose that W is an open neighborhood of the origin in C2 . We will use the coordinates x and y on W. For notational ease, when we restrict x and y to various subspaces where the domain is clear, we shall continue to write simply x and y. Let C be a complex analytic curve in W such that every component of C contains the origin. We assume that the origin is an isolated point in V (x) ∩ C and in V (y) ∩ C, i.e., that C does not have a component along the x- or y-axis. Below, we let Dǫ denote a closed disk, of radius ǫ, centered at the origin, in the complex plane. We ◦ denote the interior of Dǫ by Dǫ , and when we delete the origin, we shall superscript with an asterisk, i.e., ◦ ◦ D∗ǫ := Dǫ − {0} and D∗ǫ := Dǫ − {0}. We select 0 < ǫ2 ≪ ǫ1 ≪ 1 so that: ◦ i): the “half-open” polydisk Dǫ1 × Dǫ2 is contained in W; ◦ ii): (∂Dǫ1 × Dǫ2 ) ∩ C = ∅ (this uses that the origin is an isolated point in V (y) ∩ C) ; ◦ ◦ ◦ Note that ii) implies that (Dǫ1 × Dǫ2 ) ∩ C = (Dǫ1 × Dǫ2 ) ∩ C. ◦ ◦ ◦ y iii): Dǫ1 × D∗ǫ2 −→ D∗ǫ2 is a proper stratified submersion, where the Whitney strata are ∂Dǫ1 × D∗ǫ2 , ◦ ◦ ◦ ◦ (Dǫ1 × D∗ǫ2 ) − C, and (Dǫ1 × D∗ǫ2 ) ∩ C. ◦ ◦ y ◦ iv): (Dǫ1 × D∗ǫ2 ) ∩ C −→ D∗ǫ2 is an m-fold covering map, where m := (C · V (y))0 . ◦ ◦ ◦ Let D := (Dǫ1 × Dǫ2 ) ∩ (C ∪ V (y)). Let (x0 , y0 ) ∈ (D∗ǫ1 × D∗ǫ2 ) − D. Let σ : [0, 1] → {x0 } × Dǫ2 be a ◦ smooth, simple path such that σ(0) = (x0 , y0 ), σ(1) =: (x0 , y1 ) ∈ C, and σ([0, 1)) ⊆ ({x0 } × Dǫ2 ) − D. 4 Let S be the image of σ; as σ is simple, S is homeomorphic to [0, 1]. Let σ0 := y ◦ σ and let S0 be the ◦ image of σ0 . Thus, S0 is homeomorphic to [0, 1] and is contained in D∗ǫ2 . Lemma 2.1.(The Swing) There exists a continuous function H : [0, 1]×[0, 1] → Dǫ1 ×S0 with the following properties: a) H(t, 0) = σ(t), for all t ∈ [0, 1]; b) H(t, 1) ∈ Dǫ1 × {y0 }, for all t ∈ [0, 1]; c) H(0, u) = (x0 , y0 ) d) if H(t, u) ∈ D, then t = 1; e) H(1, u) ∈ C, for all u ∈ [0, 1]; f ) the path η given by η(u) := H(1, u) is a homeomorphism onto its image. Thus, H is a homotopy from σ to the path γ given by γ(t) := H(t, 1) ∈ Dǫ1 × {y0 }, such that (x0 , y0 ) is “fixed” and the point (x0 , y1 ) = H(1, 0) “swings up to the point” H(1, 1) by “sliding along” C, while the remainder of σ does not hit D as it “swings up” to γ. ◦ y ◦ Proof. The proper stratified submersion Dǫ1 × D∗ǫ2 −→ D∗ǫ2 is a locally trivial fibration, where the local y trivialization respects the strata. The restriction of this fibration Dǫ1 × S0 −→ S0 is a locally trivial fibration over a contractible space and, hence, is equivalent to the trivial fibration. Therefore, there exists a homeomorphism

Ψ : Dǫ1 × S0 , (Dǫ1 × S0 ) ∩ C → Dǫ1 × {y0 }, (Dǫ1 × {y0 }) ∩ C × [0, 1] such that the projection of Ψ(x, σ0 (t)) onto the [0, 1] factor is simply t, and such that Ψ(x, y0 ) = ((x, y0 ), 0). It follows that there is a path α : [0, 1] → Dǫ1 such that Ψ(σ(t)) = ((α(t), y0 ), t), for all t ∈ [0, 1]. Define H : [0, 1] × [0, 1] → Dǫ1 × S0 by
H(t, u) := Ψ−1 (α(t), y0 ), (1 − u)t . All of the given properties of H are now trivial to verify. ✷ Remark 2.2. By Property c) of Lemma 2.1, the map H yields a corresponding map H T whose domain is a triangle instead of a square. One pictures the image of H, or of H T , as a “gluing in” of this triangle into Dǫ1 × S0 in such a way that one edge of the triangle is glued diffeomorphically to S, and another edge is glued diffeomorphically onto the image of η. The third edge of the triangle is glued onto the image of γ, but not necessarily in a one-to-one fashion. 3 The Main Theorem In this section, we will prove the Main Theorem, as stated in the Introduction and as appears below as Theorem 3.1. That a) of the Main Theorem implies both b) and c) is well-known; one can, for instance, conclude 5 it from Theorem 1.1. The difficulty is to prove that b) and c) imply a). In fact, we prove the contrapositives; e n−1 (Ff ) < λ1 and dim H e n−1 (Ff ; Z/pZ) < λ1 . we prove that if we are not in the trivial case, then rank H We must first describe the machinery that goes into this part of the proof. As the value of λ1 is minimal for generic z0 , we lose no generality if we assume that our linear form z0 is chosen more generically than simply being prepolar. We choose z0 so generically that, in addition to being prepolar, the discriminant, D, of the map (z0 , f ) and the corresponding Cerf diagram, C, have the usual properties – as given, for instance, in [10], [16], and [17]. We will describe the needed properties below. e := (z0 , f ) : (U, 0) → (C2 , 0). We use the coordinates (u, v) on C2 . The critical locus ΣΨ e of Ψ e is Let Ψ e Ψ) e consists of the u-axis together with the Cerf diagram the union of Σf and Γ. The discriminant D := Ψ(Σ C := D − V (v). We assume that z0 is generic enough so that the polar curve is reduced and that, in a e | is one-to-one. neighborhood of the origin, Ψ Γ We choose real numbers ǫ, δ, and ω so that 0 < ω ≪ δ ≪ ǫ ≪ 1. Let Bǫ ⊆ Cn be a closed ball, centered ◦ ◦ at the origin, of radius ǫ. Let Dδ and Dω be open disks in C, centered at 0, of radii δ and ω, respectively. ◦ ◦ ◦ ◦ e we let Ψ One considers the map from (Dδ × Bǫ ) ∩ f −1 (Dω ) onto Dδ × Dω given by the restriction of Ψ; denote this restriction. As Bǫ is a closed ball, the map Ψ is certainly proper, but the domain has an interior stratum, and a stratum coming from the boundary of Bǫ . However, for generic z0 , all of the stratified critical points lie on Γ ∪ Σf , i.e., above D. ◦ ◦ ◦ ◦ We continue to write simply D and C, in place of D ∩ (Dδ × Dω ) and C ∩ (Dδ × Dω ). As Ψ is a ◦ ◦ proper stratified submersion above Dδ × Dω − D, and as Ψ|Γ is one-to-one, many homotopy arguments in ◦ ◦ ◦ ◦ (Dδ × Bǫ ) ∩ f −1 (Dω ) can be obtained from lifting constructions in Dδ × Dω . This is the point of considering the discriminant and Cerf diagram. ◦ ◦ Let v0 ∈ Dω − {0}. By construction, up to diffeomorphism, Ψ−1 (Dδ × {v0 }) is Ff and Ψ−1 ((0, v0 )) is Ff0 . In fact, for all u0 , where |u0 | ≪ |v0 |, Ψ−1 ((u0 , v0 )) is diffeomorphic to Ff0 ; we fix such a non-zero u0 , and let a := (u0 , v0 ). We wish to pick a distinguished basis for the vanishing cycles of f0 at the origin, as in I.1 of [1] (see, also, ◦ [4]). We do this by selecting paths in {u0 } × Dω which originate at a. We must be slightly careful in how we do this. ◦ First, fix a path p0 from a to (u0 , 0). Select paths q1 , . . . , qγ 1 from a to each of the points in ({u0 } × Dω ) ∩ C =: {y1 , . . . , yγ 1 }. The paths p0 , q1 , . . . , qγ 1 should not intersect each other and should intersect the set {(u0 , 0), y1 , . . . , yγ 1 } only at the endpoints of the paths. Moreover, when at the point a, the paths p0 , q1 , . . . , qγ 1 should be in clockwise order. Let r0 be a clockwise loop very close to p0 , from a around (u0 , 0). As we are not assuming that f had an isolated line singularity, we must perturb f|V (z0 −u0 ) slightly to have (u0 , 0) split into λ1 points, x1 , . . . , xλ1 inside the loop r0 ; each of these points corresponds to an A1 singularity in the domain. We select paths p1 , . . . , pλ1 from a to each of the points x1 , . . . , xλ1 , and paths ◦ q1 , . . . , qγ 1 from a to each of the points in ({u0 } × Dω ) ∩ C =: {y1 , . . . , yγ 1 }. We may do this in such a way that the paths p1 , . . . , pλ1 , q1 , . . . , qγ 1 are in clockwise order. The lifts of these paths via the perturbed f|V (z0 −u0 ) yield representatives of elements of Hn+1 (Bǫ , Ff0 ), e n (Ff0 ) form a distinguished basis ∆′1 , . . . , ∆′ 1 , ∆1 , . . . , ∆γ 1 . whose boundaries in H λ 6 ◦ By using the swing (Lemma 2.1), the paths q1 , . . . , qγ 1 are taken to new paths q̂1 , . . . , q̂γ 1 in Dδ × {v0 }. e n−1 (Ff0 ) is precisely Each q̂i path represents a relative homology class in Hn (Ff , Ff0 ) whose boundary in H ∆i . Theorem 1.3 follows from this. We can now prove the Main Theorem: Theorem 3.1. Suppose that dim0 Σf = 1 and dim0 Σf0 = 0. Then, the following are equivalent: a) We are in the trivial case, i.e., f has a smooth critical locus which defines a family of isolated singularities with constant Milnor number µf0 ; e n−1 (Ff ) = λ1 ; b) rank H e n−1 (Ff ; Z/pZ) = λ1 . c) there exists a prime p such that dim H e n−1 (Ff ) < λ1 , and so rank H e n (Ff ) < λ0 , and these Thus, if we are not in the trivial case, rank H inequalities hold with Z/pZ coefficients (here, p is prime). Proof. As mentioned above, that a) implies b) and c) is well-known. Assume then that we are not in the e n−1 (Ff ) < λ1 , and then indicate why the same proof applies with trivial case. We will prove that rank H Z/pZ coefficients. ◦ By Proposition 1.5, Γ 6= ∅, and so C 6= ∅. We want to construct just one new path in {u0 } × Dω , one which originates at a, ends at a point of C, and misses all of the other points of D; we want this path to ◦ swing up to a path in Dδ × {v0 }, and represent a relative homology class in Hn (Ff , Ff0 ) whose boundary is not in the span of ∆1 , . . . , ∆γ 1 . By the connectivity of the vanishing cycle intersection diagram ([5], [7]), one of the ∆′j must have a 6 0. non-zero intersection pairing with one of the ∆i , i.e., there exist i0 and j0 such that h∆i0 , ∆′j0 i = By fixing the path pj0 and all the qi paths, but reselecting the other pj , for j 6= j0 , we may assume that 6 0. j0 = 1, i.e., that h∆i0 , ∆′1 i = We follow now Chapter 3.3 of [4]. Associated to each path pj , 1 6 j 6 λ1 , is a (partial) monodromy e n−1 (Ff0 ), induced by taking a clockwise loop rj very close to pj , from a e n−1 (Ff0 ) → H automorphism Tj′ : H around xj . Let T ′ := T1′ . . . Tλ′ 1 , where composition is written in the order of [4]. We claim that T ′ (∆i0 ) is in the image of δ : Hn (Ff , Ff0 ) → Hn−1 (Ff0 ), but is not in Span{∆1 , . . . , ∆γ 1 }.  ◦ The composition r of the loops r1 , . . . , rλ1 is homotopy-equivalent, in {u0 } × Dω − {x1 , . . . , xλ1 } ∪ C , to the loop r0 (from our discussion before the theorem). By combining (concatenating) the loop r0 and the ◦ path qi0 , we obtain a path in {u0 } × Dω which is homotopy-equivalent to a simple path which swings up to ◦ a corresponding path in Dδ × {v0 }. Thus, T ′ (∆i0 ) is in the image of δ. Now, by the Corollaries to the Picard-Lefschetz Theorem in [1], p. 26, or as in [4], Formula 3.11, T ′ (∆i0 ) = ∆i0 − (−1) n(n−1) 2 h∆i0 , ∆′1 i∆′1 + β2 ∆′2 + . . . + βλ1 ∆′λ1 , for some integers β2 , . . . , βλ1 . As the ∆′1 , . . . , ∆′λ1 , ∆1 , . . . , ∆γ 1 form a basis, and as h∆i0 , ∆′1 i = 6 0, T ′ (∆i0 ) is not in Span{∆1 , . . . , ∆γ 1 }. 7 This finishes the proof over the integers. Over Z/pZ, the proof is identical, since the intersection diagram is also connected modulo p; see [5]. ✷ Remark 3.2. One must be careful in the proof above; it is tempting to try to use simply T1′ (∆i0 ) in place ◦ of T ′ (∆i0 ). The problem with this is that T1′ (∆i0 ) is not represented by a path in {u0 } × Dω − {(u0 , 0)} and, ◦ thus, there is no guaranteed swing isotopy to a corresponding path in Dδ × {v0 }. In the corollary below, we obtain a conclusion when the dimension of Σf is arbitrary. We use the notation and terminology from [11]. In particular, λsf,z (0) is the s-dimensional Lê number of f at the origin with respect to the coordinates z. Corollary 3.3. Suppose that the dimension of Σf at the origin is s, where s > 1 is arbitrary. Assume that the coordinates z := (z0 , ..., zs−1 ) are prepolar for f at the origin, and that the s-dimensional relative polar variety Γsf,z at the origin is not empty. e n−s (Ff ) and dim H e n−s (Ff ; Z/pZ) are strictly less than λs (0). Then, both rank H f,z Proof. One simply takes the codimension s − 1 linear slice N := V (z0 , . . . , zs−2 ) through the origin. Then, e n−s (Ff ) ∼ e (n−s+1)−1 (Ff ). Now, f|N has a 1-dimensional critical locus and, by iterating Theorem 1.1, H =H |N 1 s by Proposition 1.21 of [11], λf,z (0) = λf| ,zs−1 (0). The corollary nows follows at once from Theorem 3.1 N (the proof with Z/pZ coefficients is identical). ✷ As we shall see, Corollary 3.3 puts restrictions on the types of perverse sheaves that one may obtain as vanishing cycles of the shifted constant sheaf on affine space. Below, we refer to the constant sheaf on ν of ◦ ◦ dimension µν , shifted by 1 and extended by zero to all of V (f ); we write (k µν )•ν [1] for this sheaf (note that we omit the reference to the extension by zero in the notation). The isomorphisms and direct sums that we write below are in the Abelian category of perverse sheaves. ◦ µ In the trivial case, Σf consists of a single smooth component ν and φf [−1]kU• [n + 1] ∼ = (k ν )•ν [1]. Aside ◦ from the trivial case, is it possible for (k µν )•ν [1] to be a direct summand of φf [−1]kU• [n + 1]? The following corollary provides a partial answer, and generalizes the question/answer to critical loci of arbitrary dimension. Corollary 3.4. Suppose that the critical locus of f is s-dimensional, where s > 1 is arbitrary. For each ◦ s-dimensional component ν of Σf , let µν denote the Milnor number of f restricted to a generic normal slice of ν. If Σf is smooth and the generic s-dimensional relative polar variety of f is empty, then φf [−1]k • [n+1] ∼ = U ◦ (k µν )•ν [s]. If each component of Σf is smooth, and the generic s-dimensional relative polar variety of f is not empty, L ◦ then ν (k µν )•ν [s] is not a direct summand of φf [−1]kU• [n + 1]. 8 Proof. If Σf is smooth and the s-dimensional relative polar variety is empty, V (f ) has an af stratification consisting of two strata: V (f ) − Σf and Σf . As φf [−1]kU• [n + 1] is constructible with respect to any af stratification, the first statement follows. If each component of Σf is smooth, then, for generic coordinates, the s-dimensional Lê number λsf (0) P ◦ will be equal to ν µν , where we sum over s-dimensional components. Now, the second statement follows at once from Corollary 3.3, since such a direct summand would immediately imply that the dimension of e n−s (Ff ) is too big. ✷ H 4 Comments, Questions, and Counterexamples One might hope that a stronger result than Theorem 3.1 is true. For instance, given that Theorem 3.1 and Theorem 1.4 are true, it is natural to ask the following: ◦ e n−1 (Ff ) strictly less than P µν ? Question 4.1. If we are not in the trivial case, is the rank of H ν The answer to the above question is “no”. One can find examples of this in the literature, but perhaps the easiest is the following: Example 4.2. Let f := (y 2 − x3 )2 + w2 . Then, Σf has a single component ν := V (w, y 2 − x3 ), and one ◦ easily checks that µν = 1. However, as f is the suspension of (y 2 − x3 )2 , the Sebastiani-Thom Theorem (here, we need the version proved by Oka in [12]) implies e 1 (Ff ) ∼ e 0 (F(y2 −x3 )2 ) ∼ H =H = Z. Moreover, by suspending f again, one may produce an example in which f itself has a single irreducible component at the origin. Now, let α be the number of irreducible components of Σf . e n−1 (Ff ) strictly less than λ1 − α? Question 4.3. If we are not in the trivial case, is the rank of H Again, there are many examples in the literature which demonstrate that the answer to this question is “no”. One simple example is: Example 4.4. The function f = x2 y 2 + w2 has a critical locus consisting of two lines, λ1 = 2, but – using e 1 (Ff ) ∼ the Sebastiani-Thom Theorem again – we find that H = Z. However, a result such as that asked about in Question 4.3, but where α is replaced by a quantity involving the number of components of Γ, or numbers of various types of components in the Cerf diagram, seems more likely. Moreover, if we put more conditions on the intersection diagram for the vanishing cycles of f0 , we could certainly obtain sharper bounds than we do in the Main Theorem. Or, if we know more 9 topological data, such as the vertical monodromies, as in [14], we could obtain better bounds. However, other than Theorem 3.1 , we know of no nice, effectively calculable, formula which holds in all cases. Finally, Corollary 3.4 leads us to ask: Question 4.5. Which perverse sheaves can be obtained as the vanishing cycles of the constant sheaf on affine space? Unlike our previous questions, we do not know the answer to Question 4.5. 10 References [1] Arnold, V. I., Gusein-Zade, S. M., Varchenko, A. N. Singularities of Differentiable Maps II, Monodromy and Asymptotics of Integrals. Birkhäuser, 1988. [2] Caubel, C. Sur la topologie d’une famille de pinceaux de germes d’hypersurfaces complexes. PhD thesis, Université Toulouse III, 1998. [3] de Jong, Th. Some classes of line singularities. Math. Zeitschrift, 198:493–517, 1998. [4] Dimca, A. Singularities and Topology of Hypersurfaces. Universitext. Springer-Verlag, 1992. [5] Gabrielov, A. M. Bifurcations, Dynkin Diagrams, and Modality of Isolated Singularities. Funk. Anal. Pril., 8 (2):7–12, 1974. [6] Kato, M. and Matsumoto, Y. On the connectivity of the Milnor fibre of a holomorphic function at a critical point. Proc. of 1973 Tokyo manifolds conf., pages 131–136, 1973. [7] Lazzeri, F. Some Remarks on the Picard-Lefschetz Monodromy. Quelques journées singulières. Centre de Math. de l’Ecole Polytechnique, Paris, 1974. [8] Lê, D. T. . Une application d’un théorème d’A’Campo a l’equisingularité. Indag. Math, 35:403–409, 1973. [9] Lê, D. T. Calcul du Nombre de Cycles Evanouissants d’une Hypersurface Complexe. Ann. Inst. Fourier, Grenoble, 23:261–270, 1973. [10] Lê, D. T. and Perron, B. Sur la fibre de Milnor d’une singularité isolée en dimension complexe trois. C. R. Acad. Sci. Pairs Sér. A, 289:115–118, 1979. [11] Massey, D. Lê Cycles and Hypersurface Singularities, volume 1615 of Lecture Notes in Math. SpringerVerlag, 1995. [12] Oka, M. On the homotopy type of hypersurfaces defined by weighted homogeneous polynomials. Topology, 12:19–32, 1973. [13] Siersma, D. Isolated line singularities. Proc. Symp. Pure Math., 35, part 2, Arcata Singularities Conf.:485–496, 1983. [14] Siersma, D. Variation mappings on singularities with a 1-dimensional critical locus. Topology, 30:445– 469, 1991. [15] Teissier, B. Cycles évanescents, sections planes et conditions de Whitney. Astérisque, 7-8:285–362, 1973. [16] Tibăr, M. Bouquet Decomposition of the Milnor Fiber. Topology, 35:227–241, 1996. [17] Vannier, J. P. Familles à paramètre de fonctions holomorphes à ensemble singulier de dimension zéro ou un. PhD thesis, Dijon, 1987. 11