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On the vanishing cycles of a meromorphic function on the complement of its poles

We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.


Introduction
In the local setting, we consider a meromorphic function germ p x /q x : (Z, x) P 1 (C), where (Z, x) is the germ of a complex manifold and p x and q x are holomorphic germs at the point x. By definition, p ′ x /q ′ x is equal to p x /q x if and only if there exists a holomorphic germ u such that u(x) = 0 and that p x = up ′ x and q x = uq ′ x . In the global setting, a meromorphic function p/q : Z P 1 (C) is defined as the ratio of two sections, p and q, of a holomorphic line bundle L → Z over a connected compact complex manifold Z. We consider only examples of projective Z, which case insures the existence of global meromorphic functions, by Kodaira embedding theorem.
Our constructions and results in the two settings are completely similar and parallel. This is why we shall adopt in this paper a unique notation for both situations: p/q : Z P 1 (C) can alternatively mean a meromorphic germ or a global meromorphic function.
The meromorphic function induces a holomorphic function f : X → C, on the complement of X := Z \ {q = 0} of the pole locus {q = 0}. Then we call vanishing homology the relative homology H * (X, F ; Z), where F denotes a general fiber of f : X → C. Since f is a non-proper function, vanishing cycles may appear not only because of the critical points of f , but also because of a certain non-regular behavior of the fibers of f in the neighbourhood of the pole locus {q = 0}, which is more subtle to detect. We interpret the later phenomenon in terms of singularities of the meromorphic function, to which we give a precise meaning. Our study concerns the class of meromorphic functions with isolated singularities, including possible singularities occuring in the indeterminacy locus. This is a class of non-generic pencils, far beyond the class of generic pencils that is currently considered in Lefschetz theory. We send the reader to [Ti4] and [Ti3] for comments on the relations to Lefschetz theory.
While the homology of X and that of F might be very complicated, it turns out (Theorem 2.8) that the vanishing homology H * (X, F ; Z) is concentrated in dimension n = dim X and moreover that the space X/F has the homotopy type of a bouquet of spheres S n . The proof needs new technical ingredients, due to the general notion of singularity that we use here. Proposition 3.2 and Lemma 3.3 are crucial in this respect.
We prove that the polar Milnor number which we attach to an isolated G-singularity is the number of vanishing cycles which are "concentrated" at this singularity (Proposition 2.7 and Corollary 3.7). Alternatively, the polar Milnor number λ ξ at some point ξ coincides with the jump of the usual local Milnor number of the pencil of hypersurfaces. Vanishing cycles can now be detected by a multibranch jump formula (Theorem 4.1).
We also show that the global vanishing homology decomposes into a direct sum of vanishing homologies at the atypical fibres, with localization in case of G-isolated singularities (Proposition 2.7 and Corollary 3.7). We prove a general global Picard type formula for the monodromy (Proposition 5.1). Significant examples and particular cases are treated in §4.
The vanishing homology in the context of meromorphic functions extends the one of local proper holomorphic functions, which has been initiated by Milnor [Mi] and developed in many aspects ever since. The meromorphic setting also generalizes the study of the topology of polynomial functions via their singularities at infinity and the study of one-parameter families of non compact hypersurfaces, developped in the last decade. (We send the reader to [Ti3] for more details on these connections.) New phenomena may occur: unlike the polynomial case, where singularities at infinity "stay at the same place" for all fibers, in the meromorphic case singularities can split or even disappear (see Example 4.4).
The results addressed here are based on our 1999 preprint [ST2], which has not been published. Ever since, there appeared several papers which quote it, like [DN2], [GLM2], [Ti3]. The recent survey [Ti3] treats aspects of the topology of meromorphic functions on singular spaces, reviewing some of the results presented in the preprint [ST2] and in this paper. Connectivity results of Lefschetz type via nongeneric pencils (i.e. global meromorphic functions) are proved in [Ti4]. From a quite different point of view, meromorphic germs are discussed in [GLM1,2], where the main interest is the zeta-function of the monodromy. Classifications aspects have been explored by Arnold [Ar], who found the list of simple germs of meromorphic functions under natural equivalence relations.

Singularities of f along the indeterminacy locus
The definitions in this section naturally extend the ones used in case of polynomials and certain classes of regular functions on affine varieties, see [Ti1,Ti3] and also [Ti4]. According to our convention in the Introduction, we treat in parallel the local and global settings, using a common notation.
Definition 2.1. We call completed space the global hypersurface X of Z × P 1 defined by: where [t : s] ∈ P 1 . Then X is the analytic closure in Z × P 1 of the graph of f : X → C. We denote by π : X → P 1 the projection on P 1 . This is a proper holomorphic function which extends f : We endow X with a locally finite Whitney stratification W such that X is a stratum. In case of a germ at x ∈ Z of a meromorphic function one considers the germ of such a Whitney stratification at the line {x} × P 1 ⊂ X. In both situations, the local finiteness of the strata implies, by using Thom-Mather Isotopy Lemma [T]à la Verdier [Ve], the following finiteness result.
Proposition 2.2. The stratified projection π : X → P 1 with respect to W is locally topologically trivial over P 1 \ Λ f , for some finite set Λ f ⊂ P 1 . In particular, the restriction π |X = f is a locally trivial C ∞ -fibration over C \ Λ f .
We shall call the set of atypical values, and denote it by Λ f , the minimal set Λ f which satisfies Proposition 2.2. Notice that, if we take the meromorphic function q/p instead of p/q, then we get the same space X (even if X will be Z \ {p = 0} instead of Z \ {q = 0}). In particular, Λ 1 f can be identified with Λ f by the isomorphism [t : s] → [s : t].
For any subset A ⊂ P 1 , we denote X A := π −1 (A), F A := f −1 (A) and the general fibre F := F t = X ∩ f −1 (t), for some t ∈ Λ f . Let D be a small disc centered at a ∈ Λ f , such that D ∩ Λ f = {a}.
A crucial problem in investigating the topology of the fibres of f is how to detect and to control the change of topology. This is an open problem, in general, even for holomorphic germs or polynomial functions, but well understood in case the singularities are isolated.
P 1 be a germ of a meromorphic function. To every a ∈ P 1 , one associates the germ π : (X, (x, a)) → P 1 . By restriction to X = X \ (X pol ∪ X ∞ ), this defines the germ f | : (X, (x, a)) → C, which we denote by f x,a .
For some fixed x ∈ {q = p = 0}, one has a one-parameter family of germs (X, (x, a)), indexed over a ∈ P 1 \ {s = 0}. Unlike the case of holomorphic germs, here the point (x, a) is not in X but in its closure. So the germ f x,a is uniquely determined by the determination of the point a = [p(x) : q(x)].
Fourtheron, take a Whitney stratification W of X which has X as open stratum, remarking that Sing X ⊂ X pol . For all small enough radii ε of a ball B ε (x, a) ⊂ Z × P 1 centered at (x, a), the sphere S ε = ∂B ε (x, a) intersects transversally all the finitely many strata in the neighbourhood of (x, a). This defines a Milnor-Lê fibration (cf [Lê2]), i.e. a locally trivial fibration π : X D * ∩ B ε (x, a) → D * over a small enough punctured disk D * ⊂ P 1 centered at a, which restricts to a locally trivial fibration on the complement of X pol , namely: This fibration will be called the Milnor-Lê fibration of the function germ f x,a at the point (x, a) ∈ X. It depends on the point (x, a). In particular, the radius ε of the ball depends on the point a. 1 From Proposition 2.2 it follows that, since π is stratifiedtransversal to X over We now endow X with a "partial Thom stratification", cf [Ti1]. Suppose that X is endowed with a complex stratification G = {G α } α∈S such that X is a stratum and X pol is a union of strata. If G α ∩ G β = ∅ then, by definition, G α ⊂ G β and in this case we write G α < G β .
Let (x, a) ∈ X pol \ X ∞ . We consider on X the Thom (a q ) regularity condition at (x, a), see e.g. [GWPL, ch. I] for the definition. In terms of the relative conormal (see [Te], [HMS]), the condition (a q ) at ξ := (x, a) for the strata G α and G β translates to the inclusion: It is known that this condition is independent on q, up to multiplication by a unit, see e.g. [Ti1, Prop. 3.2]. We therefore may and shall refer to this as Thom regularity condition relative to X pol , at (x, a).
Definition 2.4. We say that G is a ∂τ -stratification (partial Thom stratification) relative to X pol if at any point ξ ∈ X, any two strata G α < G β with ξ ∈ G α ⊂ X pol and G β ⊂ X \ X ∞ satisfy the Thom regularity condition relative to X pol .
The Whitney stratification W of X considered in Proposition 2.2 is an example of ∂τ -stratification relative to X pol . This follows from [BMM, Théorème 4.2.1] or [Ti1, Theorem 3.9]. Nevertheless the ∂τ -stratifications are less demanding than Whitney stratifications and than Thom stratifications.
Definition 2.5. We consider the singular locus of π with respect to some ∂τ -stratification relative to X pol , denoted by G and we say that the following closed subset of X \ X ∞ : is "the singular locus of f " with respect to G. We say that f has isolated singularities with respect to G if dim Sing G f ≤ 0. We say that f has isolated singularities at a ∈ C (or at the fibre X a ) if dim X a ∩ Sing G f ≤ 0.
The space X is nonsingular and consists of one stratum. The set X ∩ Sing G f of Gsingularities on X is just the usual singular set Sing f ⊂ Z \ {q = 0}. The singularities of the new type are X pol ∩ Sing G f .
We show that singularities of this type are manageable when they are isolated. In this case one may localize the variation of topology of fibres, which phenomenon has been observed before in the case of holomorphic germs, by Milnor [Mi], and in case of polynomial and regular functions [ST1], [Ti1]. Actually the proof for meromorphic functions follows the arguments of [Ti1, Theorem 4.3] and we leave it to the reader.
For instance, when dim Z = 2, the pencil p/q has isolated G-singularities, relative to the coarsest partial Thom stratification G, if and only if the fibres of f are reduced.
Proposition 2.6. Let f have isolated singularities with respect to G at a ∈ C and let X a ∩ Sing G f = {a 1 , . . . , a k }. Then the variation of topology of the fibres of f at F a is localizable at the points a i .
The localization result implies that the vanishing cycles are concentrated at the isolated singularities, as follows: Proposition 2.7. Let f have isolated singularities with respect to G at a ∈ C and let X a ∩ Sing G f = {a 1 , . . . , a k }. Let D ⊂ C be a small enough closed disc centered at a and let s ∈ ∂D. Then, for any small enough balls B i ⊂ Z × C centered at a i , we have: Proof. Note first that in the local setting we have just one singular point, i.e. k = 1. (a). A general Lefschetz duality result (see e.g. [Br,Prop. 5.2]) says that, since we work with triangulable spaces, we have: Next, the cohomology group splits, through excision, into local contributions, by Proposition 2.6: where the second equality holds because B i ∩ X s is contractible, for small enough ball The same Lefschetz duality result may be applied locally to yield: Our main result concerning vanishing cycles at the level of homotopy type is the following.
Theorem 2.8. Let f have isolated singularities with respect to some ∂τ -stratification G relative to X pol . Let F be a general fibre of f and let D ⊂ C be a small enough open disc centered at some a ∈ Λ f . Then the space X, resp. F D , is obtained from F to which one attaches a number of cells of real dimension n = dim Z. In particular we have the following homotopy equivalences: (a) X/F ≃ S n , in the global setting.
(b) F D /F ≃ S n , in the local or global setting.
In the local setting, by F D and F we mean the intersections with some small sphere B ε , so (b) should read: We shall give the proof in §3, after introducing a few technical ingredients. The number of spheres will be discussed in Corollary 3.7. In the local setting, Theorem 2.8 extends Milnor's bouquet theorem [Mi], whereas in the global setting, it extends the bouquet result for polynomial functions [ST1, Theorem 3.1] and is of similar flavor as [Ti1, Theorem 4.6].

Polar curves and Milnor numbers at the indeterminacy locus
We show first that an isolated G-singularity at a point of X pol is detectable by the presence of a certain local polar locus, which we define in the following.
Definition 3.1. Let ξ = (x, a) ∈ X pol \X ∞ and consider a small neighbourhood V ⊂ Z of x. Let Sing f , respectively Sing (f, q), denote the singular locus of the restriction The polar locus Γ ξ (f, q) is the germ at ξ of the analytic space: From the definition we get the isomorphism Γ ξ (f, q) ≃ Γ ξ (p, q). The polar locus depends on the multiplicative unit u, i.e. Γ ξ (f, qu) is different from Γ ξ (f, q), meanwhile we shall prove that it induces well defined local invariants. If the polar locus is void, the multiplicity in (b) is zero. Suppose next that Γ ξ (f, q) has dimension 1. Consider a small enough ball B ⊂ Z × C centered at ξ, to fit in the Milnor-Lê fibration of the function π at ξ: where D ⊂ C is centered at a. The notation Γ(f, q) will stay for the representative in B of the germ Γ ξ (f, q). We may choose D so small, that for all s ∈ ∂D, those intersection points X s ∩ Γ(f, q) which tend to ξ when s tends to a, are inside B. This is possible because Γ(f, q) is a curve which cuts X pol at ξ. We shall compute the homology H * (B ∩ X s ) of the Milnor fibre of the fibration (3.1). Inside B, the restriction of the function q to B ∩ X s has a finite number of isolated singularities, which are precisely the points of intersection B ∩ X s ∩ Γ(f, q).
We claim that the space B ∩ X s ∩ q −1 (δ) is contractible, for small enough discδ ⊂ C centered at 0. To prove it, we need the following: Lemma 3.3. Let f have isolated G-singularities at ξ. Let B be a small enough ball at ξ such that the sphere S := ∂B cuts transversely all those finitely many strata of G which have ξ in their closure and does not intersect other strata.
Proof. If the statement was not true, then there would exist a sequence of points η i ∈ S ∩ (X \ X pol ) tending to a point η ∈ S ∩ X a ∩ X pol , such that the intersection of tangent spaces Assuming, without loss of generality, that the following limits exist, we get the inclusion: Let G α ⊂ X pol be the stratum containing η. Remark that dim G α ≥ 2, sinceḠ α ∋ ξ and π ⋔ η G α . This implies that dim G α ∩ X a ≥ 1. We have, by the definition of the stratification G, that lim In conclusion, the intersection in (3.2) contains T η (G α ∩ X a ). But, since S ⋔ η G α , the limit lim T η i (S ∩ X) cannot contain T η (G α ∩ X a ) and this gives a contradiction.
Letδ be so small that B ∩ X s ∩ q −1 (δ) ∩ Γ(f, q) = ∅. By the Lemma 3.3 above and by choosing appropriate D andδ, the map q : B ∩ X s ∩ q −1 (δ * ) →δ * is a locally trivial fibration. Therefore B ∩ X s ∩ q −1 (δ) is homotopy equivalent, by retraction, to the central fibre B ∩ X s ∩ X pol . This proves our claim.
We now remark that the central fibre B ∩ X s ∩ X pol is just the complex link at ξ of the space X pol . The space X pol is a product ({q = 0} ∩ {p = 0}) × C at ξ, along the projection axis C, hence its complex link is contractible, and so is B ∩ X s ∩ q −1 (δ).
Pursuing the proof of Proposition 3.2, we observe that B ∩X s is homotopy equivalent to B ∩ X s ∩ q −1 (δ), for D and δ like in Lemma 3.3 and, in addition, the radius of D much smaller than the radius of δ. This supplementary condition is meant to insure Now, the total space B ∩ X s ∩ q −1 (δ) is built up by attaching to the space B ∩ X s ∩ q −1 (δ), which is contractible, a finite number of cells of dimension n − 1, which correspond to the Milnor numbers of the isolated singularities of the function q on B ∩ X s ∩ q −1 (δ \δ). The sum of these numbers is, by definition, the intersection multiplicity mult ξ (Γ(f, q), X a ).
We have proven that: When replacing all over in our proof the function q by qu, we get the same relation (3.3), with qu instead of q. This concludes our proof of 3.2.
The above proof shows that B ∩ X s is, homotopically, a ball to which one attaches a certain number of (n − 1)-cells. We therefore get: Corollary 3.4. Let f have an isolated G-singularity at ξ. The fibre B ∩ X s of the local fibration (3.1) is homotopy equivalent to a bouquet of spheres S n−1 .
Definition 3.5. We denote the number of spheres by λ ξ := dim H n−1 (B ∩ X s ) and call it the polar Milnor number at ξ. We say that f has vanishing cycles at ξ if λ ξ > 0.
In the global setting, then we denote by λ a the sum of the polar Milnor numbers at singularities on X a ∩ X pol and also denote λ = a∈Λ f λ a .
3.1. Proof of Theorem 2.8. We take back the notations of Theorem 2.8. Since Sing G f ⊂ X \ X ∞ is a finite set of points, the variation of topology of the fibres of f is localizable at those points (cf. Proposition 2.6). Let X a ∩ Sing G f = {a 1 , . . . , a k }, with k = 1 in the local setting.
For some point a i ∈ X∩Sing G f , by Milnor's classical result for holomorphic functions with isolated singularity [Mi], it follows that the pair ( In case a i ∈ X pol ∩ Sing G f , we may invoke the following lemma, which is a version of a result by Hamm and Lê [HL, Corollary 4.2.2] for our partial Thom stratification: We conclude that the space X is built up starting from a fibre F , then moving it within a fibration with a finite number of isolated singularities. By the above connectivity result and by Switzer's result [Sw, Proposition 6.13], at each singular point one has to attach a number of cells of dimensions ≥ n. In fact the cells to be attached are of dimension precisely n, by the following reason. We may apply the duality Proposition 2.7(b) and invoke Corollary 3.4, which show that the relative homology H * (F Da , F s ) is concentrated in dimension n.
Then one can map a bouquet of n spheres into F D /F s such that this map is an isomorphism in homology. This implies, by Whitehead's theorem, that the map induces an isomorphism of homotopy groups. (Remark that F D /F s is simply connected whenever n ≥ 2). Since we work with analytic objects, therefore triangulable, the space F D /F s is a CW-complex. For CW-complexes, weak homotopy equivalence coincides with homotopy equivalence.
Let us remark that the total number of n-cells is the sum of the local Milnor numbers, resp. the polar Milnor numbers. This ends the proof of our theorem.
As consequence, we get the relative Betti numbers (see Proposition 2.7). This may be compared to similar formulas in case of polynomial functions [ST1].
Corollary 3.7. Let f have isolated G-singularities at a ∈ C with respect to some ∂τ stratification G. Then H j (F D , F s ) = 0 for j = n and: where µ a is the sum of the Milnor numbers of the singularities of F a and λ a denotes the sum of the polar Milnor numbers at X a ∩ X pol .
In particular, if f has isolated G-singularities at all fibres, then: where µ is the total Milnor number of the singularities of f on Z \ {q = 0} and λ is the total polar Milnor number at X pol \ X ∞ .

Vanishing cycles in special cases and examples
The singular locus Sing X ⊂ Z × C is contained in X pol and can be complicated. We have Sing X \ X ∞ = ∪ t∈C (Sing X t ) ∩ X pol . However, X pol \ Sing X is a Whitney stratum and Sing X is a union of Whitney strata, in the canonical Whitney stratification W of X which has X as a stratum.
We shall consider here a ∂τ -stratification G which may be coarser than W (and which exists, by Definition 2.4 and the remark following it). Then Sing G f ∩ X pol ⊂ Sing X. Indeed, this follows from the fact that the space X pol \ X ∞ is locally a product {q = p = 0} × C and the projection π is transversal to it off Sing X.
In particular, for n = 2, f has isolated G-singularities at a ∈ C if and only if F a is reduced.
Let ξ = (x, a) ∈ X pol \ X ∞ . We assume in the following that dim ξ Sing X a = 0. This implies that dim ξ Sing G f ≤ 0 and that the germ (Sing X, ξ) is either a curve or just the point ξ. If it is a curve, then it can have several branches and its intersection with X s is, say, {ξ 1 (s), . . . , ξ k (s)}, for any s ∈ D * , where D ⊂ C is a small enough disc at a.
The germs (X s , ξ i (s)) are germs of hypersurfaces with isolated singularity. Let µ i (s) denote the Milnor number of (X s , ξ i (s)). Then k i=1 µ i (s) ≤ µ(a). Equality may hold only if k = 1, by the well known non-splitting result of Lê D.T [Lê1]. In general, we have: Theorem 4.1. Let dim ξ Sing X a = 0 and dim ξ Sing X = 1. Then: Proof. The hypothesis implies that the germ of Sing G f at ξ is the point ξ or it is void. For any s ∈ D small enough, the germ (X s , ξ i (s)) is locally defined by the function: We have that, locally at ξ, the singular locus Sing h is equal to Sing X, in particular included into X pol . Consider the map (h, t) : (Z × C, ξ i (s)) → C 2 . Note that the polar locus Γ ξ (h, t) is a curve or it is void, since ξ is an isolated G-singularity. Following [Lê2], there is a fundamental system of privileged polydisc neighbourhoods of ξ in Z × C, of the form (P α × D ′ α ), where D ′ α ⊂ C is a disc at a and P α is a polydisc at x ∈ Z such that the map is a locally trivial fibration over (D * α ×D ′ α )\Im(Γ(h, t)). We chose D α and D ′ α such that is contractible, since it is the Milnor fibre of the linear function t on a smooth space. This is obtained, up to homotopy type, by attaching to (h, t) −1 (0, s) ∩ (P α × D ′ α ) a certain number r of n-cells, equal to the sum of the Milnor numbers of the function h | : t −1 (s) ∩ (P α × D ′ α ) → D α . Since we have the homotopy equivalence (h, t) −1 (0, s) ∩ (P α × D ′ α ) ≃ B ∩ X s , we get, by Corollary 3.4 and Definition 3.5, that is homotopy equivalent to the Milnor fibre of the germ (X a , ξ), which has Milnor number µ(a).
α ) by attaching exactly r cells of dimension n (coming from the polar intersections) and of a number of n-cells coming from the intersections with Sing h. This number of cells is, by definition, k i=1 µ i (s). We get the equality µ(a) = r + k i=1 µ i (s). Since r = λ ξ , our proof is done. Remark 4.2. If in the hypothesis of Theorem 4.1 the dimension of Sing X is not 1 but 0, then the result still holds, with the remark that in this case µ i (s) = 0, ∀i and ∀s ∈ D * . Hence λ ξ = µ(a).
We give in the remainder three examples, one on P 2 (C) with no singularities in the complement of X pol and two on a nonsingular quadratic surface in P 3 (C).
p,q : f = x(z a+b + x a y b ) y p z q , with a + b + 1 = p + q and a, b, p, q ≥ 1.
This defines a meromorphic function on P 2 (C). For some t ∈ C, the space X t is given by: We get the total jump λ = b + ap. A straightforward computation shows that µ = 0.
Example 4.4. Let Z ⊂ P 3 be the nonsingular hypersurface given by h = x 2 +z 2 +yw = 0 and consider the meromorphic function y/x. It has its axis {x = y = 0} tangent to h = 0 at [0 : 0 : 0 : 1]. By computations we get µ = 0, λ = 1 (jump A 0 → A 1 , at t = 0). The general fibre F is contractible, the special fibre F 0 is C ⊔ C and X is homotopy equivalent to S 2 . All the connected components of fibres are contractible, however the global vanishing homology is generated by a relative 2-cycle. We may remark here that the jump A 0 → A 1 cannot occur in case of polynomial functions at infinity. (4.2) x(z 2 − x 3 + xz 2 ) = tz 3 .
For all t, this is a D 5 singularity, so no jumps. Along [0 : 1 : 0 : 0] × C, in the chart y = 1 and, again, x and z as coordinates on X, we have the family of curves (germs of X t ): (4.3) x(z 2 + x) = tz 3 .

Monodromy fibration and a global Picard phenomenon
We continue to consider the local and global settings in the same time. LetD ⊂ C be a closed disc, big enough such thatD ⊃ Λ f \∞ , where ∞ denotes the point [1 : 0] ∈ P 1 . Let D i ⊂D be a small enough closed disc at a i ∈ Λ f , such that D i ∩ Λ f = {a i }. Take a point s on the boundary ofD and, for each i, a path γ i ⊂D from s to some fixed point s i ∈ ∂D i , with the usual conditions: the path γ i has no self intersections and does not intersect any other path γ j , except at the point s. By Proposition 2.2, the fibration f : X \ f −1 (Λ f ) → C \ Λ f is locally trivial, hence we may use excision in the pair (FD, F s ) and get an isomorphism (induced by the inclusion of pairs): In the particular case of polynomial functions, the above Picard formula (extracted from our preprint [ST2]) was independently noticed in [NN2] and [DN1].
We end by an easy consequence, remarked in the special case of polynomial functions in [DN2], which holds in our more general setting of local and global meromorphic functions.