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The vanishing neighbourhood of non-isolated singularities

We study the vanishing neighbourhood of non-isolated singularities of functions on singular spaces by associating a general linear function. We use the carrousel monodromy in order to show how to get a better control over the attaching of thimbles. For one dimensional singularities, we prove obstructions to integer (co)homology groups and to the eigenspaces of the monodromy via monodromies of nearby sections. Our standpoint allows to find, in certain cases, the structure of the Milnor fibre up to the homotopy type.


Introduction
In the landscape of singularities of holomorphic functions, the non-isolated singularities play a particular role. Their study has been initiated in the '70 and '80 by Y. Yomdin, R. Randell and Lê D.T. As a natural first extension of isolated singularities, the case of 1-dimensional singularities got special attention. The first results about the homotopy type of the Milnor fibre were proved by D. Siersma [Si1] and his studies opened the way to a series of other results for 1, 2 or higher dimensional singularities, by R. Pellikaan, T. de Jong, A. Zaharia, A. Némethi, G. Jiang, J. Fernandez and others. Meanwhile, the progress in stratified Morse theory by Goresky and MacPherson allowed one to treat singular holomorphic functions on singular spaces. The viewpoint due to Lê D.T. which consists in associating to the function f a general linear function l and studying the couple (l, f ) gave rise to new insight in the topic of non-isolated singularities. In particular, the study of the "box" neighbourhood lead to new results more recently, which exploit further the properties of the monodromies appearing in the fibration defined by (l, f ) (cf D. Siersma, J. Steenbrink, M. Saito, D. Massey and others). There are some other streams of research, which we shall not mention here.
Let f : (X, 0) → C be a holomorphic function defined on the germ (X, 0) of a singular space of pure dimension n + 1, embedded into (C m , 0) for some m. We work under the following technical but natural condition: "the rectified homotopical depth of X is maximal", which includes the cases X is smooth or a complete intersection. Our aim is to find how the Milnor fibre F of f is build from ingredients associated to restrictions of f to lower dimensional slices of the space, which have therefore lower dimensional singularities.
It is well known that the Milnor fibre F of f is (n − k)-connected, where k denotes the dimension of the stratified singular locus of f . This follows by a Lefschetz-type argument, from comparing F to the Milnor fibre F ′ of the restriction f |l=0 , where l is general enough. We go beyond this comparison and relate F to F ′ via a slice {l = η} near to the origin (Theorem 2.2): H n (F ) and H n−1 (F ) are the kernel and respectively the cokernel of a certain morphismH n (F ∪ F ′ D ) → H n (F ′ D , F ′ ), see Figure 1 for a rapid location. This comparison allows us to exploit the l-monodromy and deduce bounds for the betti numbers b n−1 (F ) and b n (F ). The pair (F ′ D , F ′ ) has the advantage that it can further be localised, by excision, in a tubular neighbourhood of the slice Sing f ∩ F ′ D of the singular locus.
We then specialise to the case dim Sing f = 1. Here we show how the l-monodromy is related to the "vertical monodromy" of the transversal singularities of a slice near to the origin. We get more explicit bounds for b n−1 (F ) and for the eigenspaces and the maximal Jordan blocks of the monodromy, as well as divisibility results for the characteristic polynomial of the monodromy h, in terms of transversal singularities. These extend to a more general setting some of Siersma's results proved in [Si2]. Our proof is based on the controlled attaching of cells, whereas Siersma's proof uses special variation maps.
We finally show how our standpoint allows to recover and throws a new light on Siersma's bouquet result for "line singularities" (Corollary 4.3). Further developments are in progress as a joint work with Dirk Siersma [ST].

A geometric viewpoint
Let (X, 0) be a germ of a singular space of pure dimension n + 1 and let f : (X, 0) → C be a holomorphic function. We shall assume that our space (X, 0) satisfies one of the following two conditions (where the former implies the latter). We refer to [HL] for the notions of rectified homotopical depth rhd(X) and rectified homological depth rHd(X).
These conditions are both true in case X is a complete intersection (arbitrarily singular). It was proved by Hamm and Lê [HL] that condition (*), resp. (**), implies that the complex link of (X, p), for any p, is homotopically equivalent to a bouquet of n-spheres, respectively has the reduced homology concentrated in dimension n. These also imply that any function with a stratified Morse singularity at some point of X has Milnor fiber homotopy equivalent to a bouquet of n-spheres, resp. with reduced cohomology concentrated in dimension n. Condition rHd(X) ≥ n + 1 is further equivalent to the fact that the constant sheaf on X is perverse.
We denote by Sing φ the singular locus of some holomorphic map φ : (X, 0) → (C p , 0), defined as the closure of the union of the singular loci Sing φ |W i of the restriction of φ to the strata W i of some Whitney stratification W of X which we fix throughout the paper.
Let us chose a generic linear function l : (X, 0) → C and let B ε denote a Milnor ball for f , that is the intersection of a small enough ball at the origin of the ambient space with a suitable representative of the germ (X, 0). Lê D.T. showed (see e.g. [Lê2]) that one can use the map (l, f ) : B → C 2 and a neighbourhood "box" B := B ε ∩ l −1 (D η ′ ) ∩ f −1 (D γ ′ ) in order to describe the local Milnor fibration of f and its relation to the Milnor fibration of the slice f |l=0 . We have already used this approach in [Ti1,Ti2] in order to get control over the attaching of cells via the monodromy, in case of a function f with isolated singularity. Let us set the notations and recall several facts.
If l is general enough, then the polar locus Γ := closure{Sing (l, f ) \ {f = 0}} is a curve on (X, 0), or it is empty. Each point of the intersections B ∩Γ∩l −1 (η) and of B ∩Γ∩f −1 (γ) is a stratified Morse singularity of the restrictions f l −1 (η) and l f −1 (γ) respectively. Let us denote by ∆ := (l, f )(Γ) the Cerf diagram in the target space C 2 . Take η ≪ η ′ and γ ≪ γ ′ such that the intersection (D η × {γ}) ∩ ∆ with the Cerf diagram ∆ is contained in the interior of D η × {γ}. We use the following notations and remarks: ( is the Milnor fiber of f . Instead of the point γ one can take any point on ∂D γ . (2) F ′ := B ∩(l, f ) −1 (η, γ) is homeomorphic to the Milnor fiber of the restriction f |{l=0} (but of course has a different monodromy over the circle {η} × ∂D η ).
, which is the complex link of the hypersurface f −1 (0) at the origin. We collect below a bunch of results from the literature, which will play a key role in the following.
(a) Under condition (*), the complex link of X at 0 is homotopy equivalent to a wedge of spheres ∨S n . The complex link of the hypersurface f −1 (0) at the origin, denoted lk(f −1 (0), 0), is homotopy equivalent to a wedge of spheres ∨S n−1 . This follows from the fact that f −1 (0) inherits the property (*), see [HL,Th. 3 The complex link of X at 0, denoted lk(X, 0), is obtained by attaching to the complex link of f −1 (0) at the origin, denoted lk(f −1 (0), 0), a number of cells, which are all of dimension n if condition (*) is assumed. Each cell corresponds to a point of intersection of the polar curve Γ with the slice l −1 (η), see [Lê1,Lê2,Ti1].
In Figure 1, this can be visualised as attaching to F ′ D the cells corresponding to the intersection Γ ∩ l −1 (η).
(c) There is a topological disk {η} × δ containing all intersection points ∆ ∩ {l = η} and is disjoint from the disk {η}×D γ . This disk can be made sliding along the Cerf diagram to a zone W ⊂ D η × {γ}. This procedure, due to Lê D.T., is known as "rabattement dans le diagramme de Cerf" and was used in our bouquet structure theorem [Ti1] for the Milnor fibre of function germs f with isolated singularity. So, revisiting the point (a) above, after attaching to F ′ D the cells from the zone (l, f ) −1 (W ), which we have identified to (l, f ) −1 ({η} × δ), we get the complex link of X, hence a bouquet of n-spheres. If we continue to attach the other n-cells coming from the intersection points of Γ with F which are outside (l, f ) −1 (W ), then, as result, we can only get more n-spheres in the bouquet.
Let h denote the monodromy on the (co)homology of the fibre of f . This is induced by a geometric monodromy which acts on F and on F ′ , and which we shall denote by the same symbol h. It acts as the identity on F D and on F ′ D . There is also the action of the geometric l-monodromy: on F D , F and F ′ D it is isotopic to the identity, but it may be non-trivial on F ′ .
Theorem 2.2. Let (X, 0) satisfy the condition (**) and let f : (X, 0) → C be any holomorphic function germ. Then the following sequences are exact: →H n (F ) → 0 and there are natural isomorphisms for all j ≥ 1: The monodromies h and l act on all these morphisms.

Proof. Let us first consider the union
Note that both the monodromies h and l are in general not isotopic to the identity on , and one can also deduce the exact number of spheres in the bouquet. Since . The same is true in cohomology.
Consider next the the pair (F ∪ F ′ D , F ). By excision we get the isomorphism: , and its similar counterpart in cohomology. The geometric monodromies h and l act on all these sequences.
Let then consider the long exact sequence of the triple (F D , F ∪ F ′ D , F ). This splits into short sequences since the homology H * (F D , F ∪F ′ D ) is concentrated in dimension n+ 1. In order to complete the proof of our complete claim we just have to remark the isomorphism H * (F D , F ) = H * −1 (F ). The proof in cohomology parallels the one in homology.
We derive the following bound for the betti number b n (F ).
Proof. This follows from the first map in the above theorem and the computation dimH n (F ∪ F ′ D ) = λ 0 +b n (lk(X, 0)) explained in Facts 2.1(b)(c). Also remember that the (co)homology of lk(X, 0) is concentrated in dimension n, since condition (**) is assumed. Note that λ 0 depends only on (X, 0) and f but not on the choice of generic l.
If X is nonsingular, then lk(X, 0) is acyclic and we recover the known well-known inequality b n (F ) ≤ λ 0 .
We further investigate the pair (F ′ D , F ′ ). Let then T denote a tubular neighbourhood of Sing f ∩F ′ D within F ′ D . Therefore T retracts to the complex link of Sing f at the origin. 1 Let L denote the action on (co)homology of the l-monodromy. We get: (a) In Theorem 2.2 one may replace H * (F ′ D , F ′ ) by H * (T, T ∩F ′ ), and similarly in cohomology.
Proof. (a) The substitution is due to the excision: . Point (b) follows from (a) and from the exact sequence of Theorem 2.2(a) on which the l-monodromy acts as follows. The action of the geometric l-monodromy on F is isotopic to the identity. It is therefore the identity on the (co)homology of F and of (F D , F ).
Remark 2.5. Since the geometric l-monodromy is the identity on F , it induces the identity on H n−j (F ′ D , F ′ ) and on H n−j (T, T ∩ F ′ ) for all j ≥ 1, by Theorem 2.2(b) and Corollary 2.4(a).

Case dim Singf = 1
We specialize to the case dim Sing f = 1 and point out several consequences of the preceding results. We still assume in this section that (X, 0) satisfies the condition (**). In case of dim Sing f = 1, the tubular neighbourhood T consists of small Milnor balls B i at the finitely many points Sing f ∩ F ′ D . Let F i denote the Milnor fiber of such an isolated singularity of the restriction of f to the transversal slice F ′ D . Then the geometric monodromy h on F restricts to the Milnor monodromy h i of F i , for each F i . Let L denote, as before, the action on (co)homology of the l-monodromy. The l-monodromy acts on the points Sing f ∩ F ′ D by certain permutations; each point comes back to itself after applying a number of times the l-monodromy. We denote by ν i the action on the (co)homology of F i of the come-back monodromy 2 . One notes that F i as well as ν i depend only on the component Σ j of Sing f . We then get: Proposition 3.1. If condition (**) holds and if dim Sing f = 1 then: where the sum is taken over the components Σ j of Sing f .

Proof. We observe that H * (T, T ∩ F ′ ) is isomorphic, by excision, to
where the sum is taken over all the singular points which are in the linear slice F ′ D . As shown in [Si3] or [Ti2], there is a cyclic movement of a singular point belonging to some component Σ j of Sing f . This yields a particular shape of the matrix of L: one can start with a basis (e) of H n−1 (F 1 ) and then (L k (e)) is a basis for H n−1 (F k+1 ), for all k ∈ 1, s j − 1, where s j is the number of points of intersection of a hyperplane slice l = η with the component Σ j of Sing f . This gives the following direct sum splitting: where in the second sum we take one point for each component Σ j of Sing f .
In particular char h|H n−1 (F ) divides the product Σ j char h j |H n−1 (F j ) .
In case of non-singular X, this result was proved by Dirk Siersma [Si2] with a different proof. A weaker version, yet for singular (X, 0) satisfying (**), was proved in [Ti2].
Proof. From Corollary 2.4(b) and Proposition 3.1 we deduce the surjection: from which (a) follows immediately. (b). The monodromy h acts on the preceding surjection. On the left hand side this amounts to the action of ⊕ j h j . One remarks that the monodromies h j and ν j commute. Then apply char h to this and get the claimed divisibility.
Let us remark that the analogous results are true in cohomology by standard reasons; one just replaces "surjection" by "injection" and "coker" by "ker".
Let b λ (V, µ) denote the dimension of the eigenspace corresponding to the eigenvalue λ of the linear operator µ acting on the vector space V . Let J λ (V, µ) denote the maximum of the sizes of the Jordan blocks. With these notations we have: Corollary 3.3. In cohomology, let K j := ker(ν j − id | H n−1 (F j )). Then: Proof. (a) is clear from the preceding remark.
(b) and (c). The monodromy h acts on the inclusion morphism (a), so h − λid acts too. On the right hand side term, this amounts to the action of h i , respectively of h i − λid, on H n−1 (F i ), independently for each i. We know that h i commutes with ν i .
The study may be pursued in case of higher dimensional singular locus Sing f ; this is work in progress jointly with Dirk Siersma [ST].

Controlled attaching
This is a study of the cell-attaching described at the end of Facts 2.1(b) in the case dim Sing f = 1 and (X, 0) = (C n+1 , 0). We refer to the notations and results in §2; let us recall that condition (*) is fulfilled in this case.
We shall use, and therefore need to recall from [Ti1] the construction of relative thimbles associated to the pair (F, F ′ ) which are adapted to the carrousel monodromy. The disk D η × {γ} will be called carrousel disk.
Our study on the attaching of thimbles starts with the following 5 steps. We refer to Figure 1. The first 3 steps apply to a singular space (X, 0) under the condition (*).
(1). Our assumptions imply that F ′ is a bouquet of n − 1 spheres. The restriction of f to the slice l = η has only isolated singularities: the intersections with Singf and the intersections with Γ (which are of Morse type, by the genericity of l). Therefore this is a deformation of the singularity f |l=0 which is a partial Morsification.
(2). The result of attaching to F ′ the thimbles corresponding to the singularities in the zone (l, f ) −1 ({η} × δ) is, up to homotopy type, a bouquet of spheres of dimension n − 1. The number of spheres is equal to the sum of the Milnor numbers of the singularities of the slice {f = 0} ∩ {l = η}. Now, remember from Facts 2.1(b) that we may identify the by attaching a number of n-cells, each cell corresponding to one of the intersection points Γ∩F . The total number of cells is equal to the intersection multiplicity mult 0 (Γ, {f = 0}). We first attach to F ′ the thimbles from the zone (l, f ) −1 (W ). We have seen before what is the result of this attaching.
If ∆ is not empty, then there exist thimbles outside the zone (l, f ) −1 (W ); this is due to the fact that all the components of ∆ are tangent to the horizontal axis {f = 0}.
Let us attach one "next" thimble, which is out of the zone (l, f ) −1 (W )). By the main construction in [Ti1], this thimble is the image by the carrousel monodromy of a thimble from the zone (l, f ) −1 (W ). Call the latter thimble t 0 and the former t 1 . Say t 0 attaches to F ′ over the cycle a. By the construction in [Ti1], the thimble t 1 will attach to F ′ exactly over the cycle h 1 (a), where h 1 denotes the geometric f -monodromy in the slice F ′ D around the singular points Sing f ∩ {l = η}. All the above explanation is already contained in [Ti1], where the case dim Sing f = 0 is treated; in that case the monodromy h 1 is geometrically trivial.
(4). What is h 1 (a) more precisely in the case (X, 0) = (C n+1 , 0) and dim Sing f = 1? We consider the cycles up to homotopy equivalence and denote by [·] the homology or homotopy equivalence classes. In particular the additive notation for homotopy classes means that our homotopy groups are restricted to dimension ≥ 2 and ≥ 3 for relative homotopy groups (i.e. thimbles) respectively.
Let us remark, firstly, that the f -monodromy in the slice F ′ D around the singular points Sing f ∩ {l = η} splits into the direct sum of the monodromies around each of these points, since they are all in the same fibre of f . Secondly, that each such singularity may be more complex than a Morse singularity and so the monodromy around each of such singular points is the Coxeter element of some Morsification of the corresponding singularity. Then, by applying the Picard-Lefschetz rules to the monodromy h 1 we get: where k r is an integer and b r denotes one of the cycles vanishing at some point of Sing f ∩ {l = η} (and where the sum is taken over these cycles).
(5). After all this discussion, we come back to the attaching of t 0 and t 1 . Let a be as chosen before. Notice now that the attaching of t 0 over a just kills a, in other words a is contractible in the space F ′ ∪ (l, f ) −1 (W ). Next, the attaching of t 1 to F ′ ∪ (l, f ) −1 (W ) is done over a cycle homotopy equivalent to [a] + r k r [b r ]. Since a is contractible in the space F ′ ∪ (l, f ) −1 (W ), this new attaching is really over the cycle r k r [b r ]. We have shown before that this cycle is not zero, and it is clearly independent on the cycles of type a. We get the following conclusion: Proposition 4.1. Let (X, 0) = (C n+1 , 0) and dim Sing f = 1. Assume that the linear function l is sufficiently general. Then F is obtained from F ′ ∪ (l, f ) −1 (W ) by attaching thimbles defined by the carrousel monodromy, with attaching maps of the type r k r [b r ], where b r 's are the cycles of F ′ vanishing at the singular points Sing f ∩ {l = η}.
An exceptional case is when no thimbles are attached to F ′ ∪ (l, f ) −1 (W ). This is equivalent to ∆ being not tangent to the axis f = 0, which can only happen if ∆ = ∅ for generic l, in which case one has F ht ≃ F ′ . This is in turn equivalent to the fact that the singular locus Singf is a line (by the non-splitting principle of Lê D.T.) and that the Milnor number of the transversal singularity is constant along the line. The equivalences can be deduced from the attaching results discussed in Facts 2.1.
Let us further remark that there exists a geometric cycle of type a (i.e. a cycle over which attaches one of the carrousel thimbles from the zone (l, f ) −1 (W )) which has nonzero intersection with at least one of the b r 's. This is due to the fact that the cycles of types a and b are together a basis of cycles (which one may arrange to be a geometric basis) of the isolated singularity at the origin of f |l=0 . Their intersection graph is connected, cf [La]. It is then an easy exercise* (which we may safely leave to the reader) to show that there exists at least one [b r ] in the sum (1) such that its coefficient k r is non-zero.
The attaching map is precisely the boundary map ∂ : H n (F, F ′ ) → H n−1 (F ′ ) in homology. In homotopy, the additive notation has a meaning only if n ≥ 3 and then the attaching map is the corresponding boundary morphism ∂ : π n (F, F ′ ; .) → π n−1 (F ′ ; .). Now remark that the monodromy h acts on the exact sequence of the pair (F, F ′ ), and its action on H n−1 (F ′ ) or π n−1 (F ′ ) is precisely h 1 . Therefore h 1 (∂(α)) = ∂(h(α)) for any linear combination α of carrousel thimbles. This proves the following, where coefficients are in some field: This corollary has been announced independently by Lê D.T. and D.B. Massey [LM]. It is necessary to compare this result to Siersma's one in [Si2] which coincides to Corollary 3.2(a) in case of a non-singular space germ (X, 0). When Singf is a union of lines, then Corollary 4.2 provides a bound which might be better by at most one. Nevertheless in all other cases Siersma's result gives a better bound, or at least the same. Proof. Let [a i ] denote the cycles of F ′ which are killed by carrousel thimbles from the zone (l, f ) −1 (W ). We claim that there exists a linear combination of carrousel thimbles [t i ] such that the attaching map to F ′ is: where s i ∈ Z. The claim follows from the following fact: if p i = a i , b denotes the intersection pairing then gcd{|p i |} i = 1. Indeed, if v := gcd{p i } i > 1 then all p i 's are zero modulo v, which contradicts the fact that the Dynkin diagram of f |l=0 relative to a distinguished basis is connected in the homology with coefficients any Z-module (see e.g. [AGV,p. 77]).
Since gcd{|p i |} i = 1, there exist integer coefficients s i such that i s i p i = 1. Therefore , which is homotopy equivalent to [b]. So this attaching will kill the cycle [b]. (b). Part (a) implies that the reduced homology of F is concentrated in dimension n. For n ≥ 3 the above attaching result holds in homotopy too. We shall treat the cases n = 1, 2 separately.
For any n ≥ 2 and in particular for n = 2, we may apply a result by Lê-Saito [LS]: this says that in our situation the fundamental group of F is abelian, hence trivial (since H 1 (F ) = 0). Then F has the homotopy groups of a bouquet ∨S n , via the Hurewicz map, and therefore, by the Whitehead theorem for CW-complexes, it is homotopy equivalent to ∨S n . As for the case n = 1, F has the homotopy type of a connected 1-dimensional CW-complex, hence F ht ≃ ∨S 1 . Corollary 4.3(b) recovers, with a different proof, Siersma's bouquet result [Si1].