ON THE MONODROMY OF COMPLEX POLYNOMIALS

by Alexandru Dimca and András Némethi

1. Introduction

Consider a polynomial function f : ℂ^{n} → ℂ with generic fiber F. Let B_{f} be the bifurcation set of f, hence f induces a smooth locally trivial fibration over ℂ \ B_{f}, see (2.1). Then, for any integer q ≥ 0 and any coefficient ring R there is an associated monodromy representation

ρ(f)_{q} : π_{1}(ℂ \ B_{f}, pt) → Aut(H̃_{q}(F, R))

in (reduced) homology as well as a monodromy representation in (reduced) cohomology

ρ(f)^{q} : π_{1}(ℂ \ B_{f}, pt) → Aut(H̃^{q}(F, R)).

Going around a circle in ℂ large enough to contain all of the bifurcation set gives rise to the so called monodromy operators at infinity, which we denote by M_{∞}(f)_{q} and M_{∞}(f)^{q} respectively.

Our main result is the following.

Theorem A.

The following statements are equivalent.

(i) M_{∞}(f)_{q} = Id; (ii) the homology representation ρ(f)_{q} is trivial.

If we assume that R is a field or that R = ℤ and the integral (co)homology of F is torsion free then (i) and (ii) are equivalent to the following: (iii) M_{∞}(f)^{q} = Id; (iv) the cohomology representation ρ(f)^{q} is trivial.

This result was first obtained in the case n = 2 by the first author [D2], using specific properties of this low dimensional situation plus some Hodge theory. The same case n = 2 was also proved by Pi. Cassou-Noguès and E. Artal-Bartolo [AC] using the splice diagrams.

Here we prove it in full generality: actually we obtain a much finer result, namely that the monodromy at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation, see Theorem (3.1). The role played by this decomposition is crucial, since there are examples of polynomials ℂ^{2} → ℂ having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan normal form, see the polynomials f_{B} and f_{C} in [D2].

Theorem (3.1) is very unexpected since in many questions it is much easier to deal with the monodromy at infinity rather than with the whole of the monodromy representation. For instance, to compute the monodromy at infinity under the Thom-Sebastiani construction is not so difficult, see [N1], [N2], but to treat the corresponding monodromy representations is much harder, see [DN].

On the other hand, a lot of information on the monodromy at infinity is known, at least in some special cases, see for instance [GN], and this can be hopefully used in conjunction with our Theorem (3.1) to get information on the monodromy representation itself.

As a by-product of Theorem A. we give conditions under which the generic fiber of the polynomial f is acyclic in a certain degree q, generalizing in this way Corollary 2 from [D2]. Of course, in the case n > 2, the Abhyankar-Moh-Suzuki Theorem is missing, so we cannot deduce from the acyclicity of the fibers that the polynomial f is equivalent to a linear form, see Zaidenberg [Z] for such examples of polynomials.

Corollary B.

Assume that R = ℤ and q is a fixed nonnegative integer. Then the following are equivalent:

(i) M_{∞}(f)_{q} = Id and H̃_{c}^{2n−q−1}(F_{b}, R) = 0 for all b ∈ B_{f}; (ii) H̃_{q}(F, R) = 0 and Ker(M_{∞}(f)_{q−1} − Id) = 0.

Here we use the notation F_{b} = f^{−1}(b) for any b ∈ ℂ. For an m-dimensional affine variety Y, the reduced cohomology with compact supports H̃_{c}^{j}(Y, R) is defined to be just H_{c}^{j}(Y, R) when j ≠ 2m and H̃_{c}^{2m}(Y, R) = Coker(p^{∗} : H_{c}^{2m}(ℂ^{m}, R) → H_{c}^{2m}(Y, R)), where p : Y → ℂ^{m} is any finite morphism.

To simplify the notation we omit from now on the coefficient ring R when it plays no special role.

To give more geometric meaning to the cohomology with compact supports, note that if S_{b} is the singular part of the fiber F_{b} and d = dim(S_{b}), then the natural morphism

H_{c}^{j}(F_{b} \ S_{b}) → H_{c}^{j}(F_{b})

is surjective for j = 2d + 1 and an isomorphism for j > 2d + 1. In particular for R a field: (i) dimH_{c}^{2n−2}(F_{b}) is equal to the number of irreducible components of F_{b} as in [NN1]; (ii) dimH_{c}^{2n−3}(F_{b}) = dimH_{1}(F_{b} \ S_{b}) if d < n − 2.

2. A Picard formula for some monodromy operators

(2.1) Let f : ℂ^{n} → ℂ be a polynomial function. It is well known that there is a (minimal) finite bifurcation set B_{f} in ℂ such that f is a C^{∞}-locally trivial fibration over ℂ \ B_{f}. If c_{0} ∈ ℂ is not in B_{f}, then F = f^{−1}(c_{0}) is called the generic fiber of f.

The fundamental group π_{1}(ℂ \ B_{f}, c_{0}) is a free group with t = |B_{f}| generators. Note that in this paper we use the following convention: the product loop a · b is obtained by going first along the loop b and then along the loop a.

Set B_{f} = {b_{1}, ..., b_{t}}. Then a convenient way to construct a set of free generators γ_{i} for this group is the following. First fix some paths p_{i} : [0, 1] → ℂ, i = 1, ..., t such that we have

(i) p_{i}(0) = c_{0}, p_{i}(1) = b_{i};

(ii) (image p_{i})∩ (image p_{j}) = {c_{0}} for any i ≠ j;

(iii) the paths p_{1},...,p_{t} leave c_{0} in the counterclockwise order.

(2.2) Definition. A family of paths (p_{i})_{i=1,...,t} with the properties (i), (ii) and (iii) is called a star with base point c_{0} and end points b_{1},...,b_{t}.

Once we fix a star as above, we have automatically a base (γ_{i})_{i=1,...,t} of π_{1}(ℂ\B_{f}, c_{0}). Indeed, consider small discs D_{i} centered at b_{i} such that (image p_{i}) ∩ D_{i} is a topological radius in D_{i}. Set (image p_{i}) ∩ ∂D_{i} = c_{i} = p_{i}(t_{i}). Then p̃_{i} : [0, t_{i}] → ℂ, p̃_{i}(t) = p_{i}(t) connects c_{0} to c_{i}. Let ∂D_{i} be the loop with both end points c_{i} and going once around b_{i} counterclockwise.

Then γ_{i} = (p̃_{i})^{−1} ∘ ∂D_{i} ∘ p̃_{i} for i = 1, ..., t are loops based at c_{0} and their homotopy classes (still denoted by γ_{i}) generate the group π_{1}(ℂ \ B_{f}, c_{0}). We emphasize that the ordered set of homotopy classes γ_{1}, ..., γ_{t} is canonically associated with the star p_{1}, ...., p_{t} and it is independent of any other choices.

(2.3) The sum decomposition of H̃_{q}(F) associated with a star.

Consider as above a polynomial f : ℂ^{n} → ℂ with a bifurcation set B_{f}, a point c_{0} not in B_{f} and a star with base point c_{0} and end points in B_{f}. This provides a direct sum decomposition of H̃_{q}(F), see [B], [DN], [NN1], [ST1], [ST2] for various degrees of generality.

Indeed, first we have the obvious isomorphism ∂^{−1} : H̃_{q}(F) → H_{q+1}(ℂ^{n}, F), next we can replace the second group by H_{q+1}(f^{−1}(∪_{i}p_{i}([0, 1]) ∪ ∪_{i}D_{i}), F) using a deformation retract and finally by excision we may replace the last group by ⊕_{i}H_{q+1}(f^{−1}(D_{i}), F_{c_{i}}). Next we can define a subgroup of vanishing cycles V_{q,i} in H̃_{q}(F) for i = 1, ..., t by pushing the subgroup ∂H_{q+1}(f^{−1}(D_{i}), F_{c_{i}}) in H̃_{q}(F_{c_{i}}) to H̃_{q}(F) along the path p_{i}.

This sum decomposition

(∗_{q})  H̃_{q}(F) = ⊕_{i=1}^{t} V_{q,i}

depends essentially on the choice of the star.

If R → K is a ring extension with K = R or K a field, then by tensor product by K over R in (∗_{q}) we get the corresponding direct sum decomposition of H̃_{q}(F, K) which will be denoted sometimes by (∗_{q}(K)).

Some notation: let m_{j} = ρ(f)_{q}(γ_{j}) = H̃_{q}(T_{j}) ∈ Aut(H̃_{q}(F)) and m_{j}^{∗} = ρ(f)^{q}(γ_{j}) = H̃^{q}(T_{j})^{−1} ∈ Aut(H̃^{q}(F)) be the monodromy operators in homology and in cohomology associated to a star as above. Here T_{j} : F → F is the homeomorphism obtained by parallel transport along the path γ_{j}.

It is clear that these operators completely determine the corresponding monodromy representation and that m_{k} · ... · m_{1} = M_{∞}(f)_{q} and m_{k}^{∗} · ... · m_{1}^{∗} = M_{∞}(f)^{q}

When we need to keep track of the degree in which these operators act, we write m_{q,j} instead of m_{j} and m_{j}^{q} instead of m_{j}^{∗}.

The following result provides key information on these monodromy operators in homology (exactly as in the case of isolated hypersurface singularities and/or the classical theory of Lefschetz pencils) and can be found in the global setting in various recent preprints: [DN], Lemma (2.4), [NN1], Theorem (1.4), [NN2], Theorem (3.1) and [ST2], Proposition (2.2). The proof of it being very simple, we reproduce it below.

(2.4) Lemma. For any cycle [c] ∈ H̃_{q}(F):

(m_{j}([c]) − [c]) ∈ V_{q,j}.

Proof. The geometric monodromy along the path γ_{j} is defined as follows. Take a qcycle c and move it along the path γ_{j} to get a cycle m_{geom}(c) in F. This means that there is a (q + 1)-cycle d such that ∂d = m_{geom}(c) − c and f(d) ⊂ γ_{j}([0, 1]). But then m_{j}([c]) − [c] = ∂[d] ∈ V_{q,j}.

(2.5) The above Lemma shows that in the sum-decomposition ⊕_{i=1}^{t} V_{q,i}, the monodromy m_{j} = ρ_{f}(γ_{j}) (1 ≤ j ≤ t) has a block-decomposition of the following form:

(∗∗_{q}) m_{j} = ( \begin{matrix}1 \\ . & . & . \\ m_{j1} & ⋯ & m_{jj} & m_{j,j+1} & ⋯ & m_{jt} \\ & & & 1 \\ . & . & . \\ & & & & & 1 \end{matrix} ) ,

where all the entries, excepting the diagonal and the j-th line, are trivial. Here m_{ji} : V_{q,i} → V_{q,j} is given by m_{ji}(a_{i}) = m_{j}(a_{i}) − a_{i} (for i ≠ j); and m_{jj} : V_{q,j} → V_{q,j} is m_{jj}(a_{j}) = m_{j}(a_{j}).

(2.6) Remarks. (i) Lemma (2.4) is a weak global affine Picard-Lefschetz formula (corresponding to the result of Picard) and it says that m_{j}([c]) − [c] can be localized in f^{−1}(D_{j}).

(ii) We have similar decompositions (∗_{q}) and (∗∗_{q}) in the more general case of a regular function f : X → ℂ where X is an algebraic varity with H_{q+1}(X) = H̃_{q}(X) = 0. Hence all our results below hold in this more general setting.

(iii) Note that in some cases the decomposition (∗_{q}) is completely determined by the individual monodromy operators m_{j} or m_{j}^{∗}, and this may be a first step in the difficult task of fitting together these operators in order to obtain the monodromy representation, see for instance [DN] where this idea is applied to the Briançon polynomial.

3. The monodromy at infinity determines the monodromy representation

In this section we prove the following result which implies Theorem A.

Theorem (3.1)

Let R = ℤ and fix a non-negative integer q and an extension ℤ → K. Then:

(i) The monodromy operator at infinity M_{∞}(f)_{q} and the decomposition (∗_{q}) completely determines the monodromy representation ρ(f)_{q}. (ii) If v ∈ H̃_{q}(F, K) has components (v_{1}, ..., v_{t}) with respect to the decomposition (∗_{q}(K)) then

M_{∞}(f)_{q}(v) = av

for some a ∈ K if and only if

m_{k} · ... · m_{1}(v) = (av_{1}, ..., av_{k}, v_{k+1}, ..., v_{t})

for all k = 1, 2, ..., t. In particular

Ker(M_{∞}(f)_{q} − 1) = H̃_{q}(F)^{inv}

the invariant homology under the monodromy representation.

The precise meaning of (i) above is that if we know the operator M_{∞}(f)_{q} as a matrix of blocks of type (∗∗_{q}), then this gives the matrices corresponding to all the operators m_{k}, k = 1, ..., t.

Before giving the proof we would like to emphasize some points.

(a) The first claim (i) holds for the cohomology monodromy representation as well when H_{∗}(F, R) is torsion free. Then M_{∞}(f)_{q} is the inverse transpose of M_{∞}(f)^{q} in the following sense. The decomposition (∗_{q}) induces a similar decomposition for H̃^{q}(F), see [DN], (8.2), which we denote by (∗^{q}). If the matrix A of M_{∞}(f)^{q} is known with respect to this decomposition (∗^{q}), then the matrix of M_{∞}(f)_{q} with respect to the decomposition (∗_{q}) is just the inverse transpose matrix (A^{t})^{−1}.

(b) The second part of (3.1) is definitely false for cohomology, e.g. for R a field, there are examples when

dimKer(M_{∞}(f)^{q} − 1) > dimH̃^{q}(F)^{inv}

see for instance the Briançon polynomial in [D2].

More precisely we have the following situation when R is a field. For a single monodromy operator we have an obvious equality

dimKer(m_{k} − 1) = dimKer(m_{k}^{∗} − 1)

and various formulas for this dimension have been obtained in [ACD] and, in full generality, in [NN1]. Moreover the vector spaces (Ker(m_{k}^{∗} − 1))_{k} are in general position in H̃^{q}(F), see [ACD] and [NN1]. This implies the following inequality (which is strict in general):

dimH̃_{q}(F)^{inv} = dimKer(M_{∞}(f)_{q} − 1) = dimKer(M_{∞}(f)^{q} − 1) ≥ dimH̃^{q}(F)^{inv}.

This discussion shows that homology and cohomology monodromy representations have each their advantages and hence it is usefull to study and use them both.

(3.2) PROOF of Theorem (3.1)

The shape of the matrices in (∗∗_{q}) implies the following obvious and equivalent facts. For any k = 1, ..., t we have

(·) The operator m_{k} is determined by the composition p_{k}·m_{k} where p_{k} : H̃_{q}(F) → V_{q,k} is the projection onto the k-th factor in (∗_{q}).

(··) m_{k}(v_{1}, ..., v_{t}) = (v_{1}, ..., v_{k−1}, v′_{k}, v_{k+1}, ..., v_{t}), i.e. the operator m_{k} acts only on the k-th component of any vector v.

The formula

(∗ ∗ ∗) m′′ · m_{k} · m′ = M_{∞}(f)_{q}

where m′ = m_{k−1} · ... · m_{1} and m′′ = m_{t} · ... · m_{k+1} implies that

p_{k} · m_{k} = p_{k} · M_{∞}(f)_{q} · m^{′−1}.

Indeed, the operator m′′ leaves the first k components of a vector v unchanged. But this shows that, if m_{1},...,m_{k−1} are already determined by M_{∞}(f)_{q}, then the same holds for m_{k}, i.e. (3.1) (i) follows by induction on k.

To get the second part (ii), we use again the decomposition (∗ ∗ ∗) and the property of m′′, to get that M_{∞}(f)_{q}(v) = av implies

m_{k} · m′(v) = (av_{1}, ..., av_{k}, v_{k+1}, ..., v_{t}).

Note for this that m_{k} · m′ leaves unchanged the last (t − k) components of any vector.

(3.3) Proof of Corollary B.

This proof uses our Theorem A and the following exact sequence from [NN1], Theorem 1.4.

(E) : 0 → Im(m_{q,j} − 1) → V_{q,j} → H̃_{c}^{2n−q−1}(F_{b_{j}}) → Ker(m_{q−1,j} − 1) → 0.

In fact, in loc.cit the group H̃_{c}^{2n−q−1}(F_{b_{j}}) is replaced by the group H^{2n−q−1}(Y, ∂Y) where Y (resp. ∂Y) is the intersection of the special fiber F_{b_{j}} with a very large closed ball B in ℂ^{n} (resp. with the boundary of this ball). The two groups are clearly isomorphic and we prefer to work with the cohomology with compact supports.

To end the proof, note that the condition (i) (resp. (ii)) in Corollary B. is equivalent to the vanishing of the first and the third (resp. the second and the fourth) terms in the exact sequence (E).

4. Some examples

(4.1) Example (non-trivial monodromy on torsion in homology)

Here we give an example of a polynomial whose generic fiber F has torsion in integral homology and the action of the monodromy on this torsion is non-trivial. Hence here we take R = ℤ.

Let Q be a three-cuspidal quartic curve in the projective plane ℙ^{2}. In an appropriate coordinate system on ℙ^{2}, the curve Q is the zero set of the polynomial

f(x, y, z) = x^{2}y^{2} + y^{2}z^{2} + x^{2}z^{2} − 2xyz(x + y + z).

It is well known that the fundamental group π_{1}(ℙ^{2} \ Q) is a finite non-abelian group of order 12, see for instance [D1], p.131.

Regarding now f as a polynomial function ℂ^{3} → ℂ we have

(i) t = 1, b_{1} = 0;

(ii) the only monodromy homeomorphism h : F → F can be represented by

h(x, y, z) = (ix, iy, iz)

where i^{2} = −1, hence h^{4} = 1.

(iii) If < h > denotes the cyclic group of order 4 spanned by h, then the quotient space F/ < h > is just ℙ^{2} \ Q. This implies that H_{1}(F) = π_{1}(F) = ℤ/3ℤ, see also Corollary (4.14) in [D1], p.133.

In the exact sequence (E) we have in this case: V_{1,1} = H_{1}(F) = ℤ/3ℤ, H̃_{c}^{4}(F_{0}) = 0 since F_{0} is irreducible and hence Im(m_{1,1} − 1) = ℤ/3ℤ. In other words, the monodromy operator in homology satisfies

m_{1,1} = M_{∞}(f)_{1} ≠ 1

On the other hand H^{1}(F) = 0 and so M_{∞}(f)^{1} = 1. This example explains our restrictions on torsion in Theorem A.

The fact that H_{1}(F) is torsion was already remarked by D. Siersma in [S], Example 7.4. The methods developed in his paper can also be used to show that in this case m_{1,1} = M_{∞}(f)_{1} ≠ 1.

(4.2) Example (trivial monodromy versus trivial intersection form)

It was shown by Neumann and Norbury [NN2] that the total space of the fibration f : f^{−1}(S_{r}^{1}) → S_{r}^{1} for r >> 0 which gives the monodromy at infinity for the polynomial f can be embedded in a natural way as an open subset of a large sphere S^{2n−1} in ℂ^{n}.

This gives a Seifert form L : H̃_{n−1}(F) × H̃_{n−1}(F) → ℤ defined in the usual way as well as the usual relation

L(1 − M_{∞}(f)_{n−1}) = S

where S is the intersection form on H̃_{n−1}(F). This equality gives the following.

(4.3) M_{∞}(f)_{n−1} = 1 implies S = 0, i.e. S(x, y) = 0 for any x, y ∈ H̃_{n−1}(F).

The converse implication is false in general, since the Seifert form L can be degenerate. For instance the polynomial f = x^{2}y^{2} + y : ℂ^{2} → ℂ has M_{∞}(f)_{1} ≠ 1 and S = 0 since F is homeomorphic to ℂ \ {0, 1}, see Bailly-Maitre [BM] for details.

This converse implication is nevertheless true for the M-tame polynomials introduced by the second author and A. Zaharia in [NZ1], as a generalization of the tame polynomials introduced by Broughton in [B].

In fact, it was shown in [NZ2] that for an M-tame polynomial the fibration f : f^{−1}(S_{r}^{1}) → S_{r}^{1} for r >> 0 which gives the monodromy at infinity for the polynomial f is equivalent to the Milnor fibration at infinity ϕ : S_{r}^{2n−1} \ f^{−1}(0) → S^{1}, ϕ(x) = f(x)/|f(x)|. This implies in the standard way that in this case the Seifert form is non-degenerate.

There is an analogy between our Example (4.2) and the relations between Ker(M(f)_{∗}− 1) and KerS in the case of a local non-isolated hypersurface singularity as discussed in detail by D. Siersma in [S], section 7. Here M(f)_{∗} stands for the corresponding local monodromy operators.

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Laboratoire de Mathématiques Pures de Bordeaux Université Bordeaux I 33405 Talence Cedex, FRANCE email: dimca@math.u-bordeaux.fr

Department of Mathematics, Ohio State University, Columbus, Ohio 43210, USA email: nemethi@math.ohio-state.edu