On the monodromy representation of polynomial maps in n variables

by András Némethi and István Sigray

Abstract

For a non-constant polynomial map f : ℂ^{n} → ℂ^{n−1} we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [ACD] or [NN], where the case n = 2 is solved. As applications, we verify the Jacobian conjecture for (f, g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [Fr], [H], [LW], [NN2] and [R], where the case n = 2 is treated.

1. Introduction

Let f : ℂ^{n} → ℂ^{k} be a complex polynomial map whose derivative at a generic point has rank k. Set F_{a} = f^{−1}(a). It follows from the general theory (and it is not difficult to prove) that there exists a hypersurface Λ ⊂ ℂ^{k} (called the bifurcation set of f) such that f is a C^{∞} locally trivial fibration over ℂ^{k} \ Λ (cf. with [Th] or [P]). In general, Λ contains the singular values, but it may contain some other values as well. For any fixed a_{0} ∉ Λ, consider the monodromy representation

ρ_{Λ} : π_{1}(ℂ^{k} \ Λ, a_{0}) → Aut H^{n−k}(F_{a_{0}}, ℂ),

whose isomorphism class is independent on the choice of a_{0}.

The monodromy representation ρ_{Λ} is one of the most important and most complex invariants of the map f. Its understanding is crucial in many different problems related with polynomial maps (e.g. in the Jacobian Problem; see also section 3 below).

Sometimes it is convenient to replace Λ by another hypersurface Λ′ of ℂ^{k} with Λ′ ⊃ Λ. In this case ρ_{Λ′} = ρ_{Λ} ∘ i_{Λ,Λ′}, where i_{Λ,Λ′} is the natural epimorphism π_{1}(ℂ \ Λ′, ∗) → π_{1}(ℂ \ Λ, ∗). In particular, the invariant subspaces of ρ_{Λ} and ρ_{Λ′} can be canonically identified. This space of invariant classes will be denoted by H^{n−k}(F_{a_{0}}, ℂ)^{ρ}. Our goal is to determine its dimension and to give a natural basis of it, provided that k = n − 1.

Notice that the formula for dim_{ℂ} H^{1}(F_{a_{0}}, ℂ)^{ρ} is known in the case (n, k) = (2, 1), see e.g. [ACD] or [NN]. Some of the existing proofs for this case are based on a detailed analysis of the global topological behaviour of polynomial functions, including their properties at infinity. The present article has a double goal: first, we generalize this low-dimensional case to arbitrary n (and k = n − 1). Our proof also shows that the statement of the theorem is rather independent of the behaviour of f at infinity. Moreover, we provide a basis for H^{1}(F_{a_{0}}, ℂ)^{ρ} as well.

Now we formulate the result. We assume that f : ℂ^{n} → ℂ^{n−1} is a polynomial map which can be written as f = p ∘ g with polynomial maps g : ℂ^{n} → ℂ^{n−1} and p : ℂ^{n−1} → ℂ^{n−1} such that the generic fiber of g is connected. (Notice that this assumption in the case n = 2 is automatically satisfied, thanks to the “affine Stein factorization property”, see e.g. [Fu] or [AB], page 100. The authors do not know if in higher dimensions this kind of factorization automatically exists or not. We thank to Zbigniew Jelonek indicating the subtility of the above factorization problem for n > 2.)

We fix a bifurcation set Λ_{g} of the polynomial g with the properties described above. Let Λ_{g} = Λ_{1} ∪ ⋯ ∪ Λ_{k} be its irreducible decomposition. Moreover, assume that for each 1 ≤ i ≤ k the hypersurface g^{−1}(Λ_{i}) has n_{i} irreducible components, each defined by the equation g_{ij} = 0 for j = 1, …, n_{i}.

Theorem. Identify H^{1}(F_{a_{0}}, ℂ) with the de Rham cohomology of F_{a_{0}} (where a_{0} ∉ Λ is fixed). Then

dim_{ℂ} H^{1}(F_{a_{0}}, ℂ)^{ρ} = ∑_{i=1}^{k}(n_{i} − 1).

Moreover, the cohomology class of the restrictions to F_{a_{0}} of the differentials

{dg_{ij}/g_{ij} : 1 ≤ i ≤ k, 1 ≤ j ≤ n_{i} − 1}  (1)

form a basis for H^{1}(F_{a_{0}}, ℂ)^{ρ}.

It is easy to see that the set (1) provides invariant classes. Indeed, take a bifurcation set Λ of f which contains p(Λ_{g}). Then notice that all the holomorphic 1-forms from (1) are restrictions of global 1-forms defined on f^{−1}(ℂ^{n−1} \ Λ).

The theorem already shows that the above construction of the dimension of the invariant subspace, and the basis (1) is independent of the choice of the set Λ_{g}. This fact is clear from the following discussion as well. Notice that if for some 1 ≤ j ≤ k and generic point a ∈ Λ_{j}, the fiber g^{−1}(a) is irreducible, then g^{−1}(Λ_{j}) is irreducible as well, i.e. n_{j} = 1 (cf. [Sh], page 77). In particular, Λ_{j} provides no contribution in the above theorem. Therefore, only those irreducible components Λ_{i} are relevant for which g^{−1}(a) is not irreducible for generic a ∈ Λ_{i}. But all these components are contained in any Λ_{g}, independently of the choice of Λ_{g}.

The proof of Theorem has two parts. First we verify the statement in the case when f = g (i.e. when the generic fiber of f is connected), then we reduce the general situation to this case using Frobenius Reciprocity property of the induced representations.

In the last section we present two applications. Both are particular cases of the famous Jacobian conjecture. More precisely, we consider two polynomial maps f : ℂ^{n} → ℂ^{n−1} and g : ℂ^{n} → ℂ such that F_{a} := f^{−1}(a) is connected for any a, and the polynomial map (f, g) : ℂ^{n} → ℂ^{n} has a nonzero constant Jacobian. In (3.1) we prove that if F_{a} is rational then (f, g) is invertible; (3.2) says that F_{a} cannot be elliptic.

These corollaries extend their analogs proved for n = 2 (see e.g. [Fr], [H], [LW], [NN2] and [R]) to arbitrary n.

2. The proof

First we assume that f = g, i.e. the generic fiber of f is connected.

Denote the coordinates of ℂ^{n} by (x_{1}, …, x_{n}), and let L_{c} be the hyperplane x_{n} = c in ℂ^{n}. By a possible change of coordinates, we may assume that there exists c ∈ ℂ such that the restriction h : L_{c} → ℂ^{n−1} of f to L_{c} = ℂ^{n−1} has a non-zero Jacobian determinant at the generic point. As we mentioned in the introduction, the statement of the theorem is independent of the choice of the bifurcation set: if for some reason we would like to increase it, we can do it freely. In the present situation, we prefer to take a bifurcation set Λ with the following additional property: h : h^{−1}(ℂ^{n−1} \ Λ) → ℂ^{n−1} \ Λ is a topological (regular) covering. Let deg(h) be its degree.

Then the statement of Theorem basically follows from the following fact.

(2.1) Proposition. One has the following exact sequence:

0 → H^{1}(ℂ^{n−1} \ Λ, ℂ) →^{f^{∗}} H^{1}(f^{−1}(ℂ^{n−1} \ Λ), ℂ) →^{r} H^{1}(F_{a_{0}}, ℂ)^{ρ} → 0,

where r denotes the restriction map.

Proof. Write B := ℂ^{n−1} \ Λ, and consider the cohomological Leray-Serre spectral sequence associated with the fibration f : f^{−1}(B) → B with fiber F_{a_{0}} (see e.g. [McC]). Then E_{2}^{p,q} = H^{p}(B, H^{q}), and the spectral sequence converges to the cohomology groups of f^{−1}(B). The stalk of H^{q} is H^{q}(F_{a_{0}}, ℂ), hence H^{0} is the constant sheaf ℂ along B, and H^{q} = 0 for q ≥ 2. Notice that E_{2}^{0,1} = H^{1}(F_{a_{0}}, ℂ)^{ρ}, hence we have the exact sequence:

0 → H^{1}(B, ℂ) →^{f_{1}^{∗}} H^{1}(f^{−1}(B), ℂ) →^{r} H^{1}(F_{a_{0}}, ℂ)^{ρ} →^{d_{2}} H^{2}(B, ℂ) →^{f_{2}^{∗}} H^{2}(f^{−1}(B), ℂ).

Next we show that d_{2} = 0; or, equivalently, f_{2}^{∗} is injective. This last fact is equivalent to the surjectivity of f_{∗} : H_{2}(f^{−1}(B), ℂ) → H_{2}(B, ℂ). This can be proved as follows. Take an arbitrary 2-cycle c in B, and consider h^{−1}(c) supported in L_{c} ∩ f^{−1}(B) ⊂ f^{−1}(B). Then f_{∗}([h^{−1}(c)]) = deg(h) · [c]. Hence the surjectivity follows. [In fact, the interested reader can verify, by generalizing slightly our covering argument, that d_{2} is zero for any Λ, even if it does not satisfy our additional assumption.] □

On the other hand, the group H^{1}(B, ℂ), respectively H^{1}(f^{−1}(B), ℂ), can be described easily. The following fact is well-known (or can be verified easily by the reader).

(2.2) Topological Lemma. Let D be a hypersurface in ℂ^{m} with irreducible components D = ∪_{i=1}^{r}D_{i}. Assume that D_{i} = {g_{i} = 0} for some irreducible polynomial g_{i} ∈ ℂ[x_{1}, …, x_{m}]. Then the cohomology classes of the forms {dg_{i}/g_{i}}_{i=1}^{r} freely generate H_{dR}^{1}(ℂ^{m} \ D, ℂ).

First, we will apply the above result for D = Λ. Fix some irreducible polynomials h_{i} such that {h_{i} = 0} = Λ_{i} (1 ≤ i ≤ k). Then the collection of classes [dh_{i}/h_{i}]_{i=1}^{k} generate freely H^{1}(B, ℂ). Next, clearly, each h_{i} ∘ f can be written as

h_{i} ∘ f = c_{i} · ∏_{j=1}^{n_{i}} g_{ij}^{α_{ij}}

for some non-zero constant c_{i}, positive integers α_{ij}, and irreducible polynomials g_{ij}. Then the classes {[dg_{ij}/g_{ij}]} (1 ≤ i ≤ k, 1 ≤ j ≤ n_{i}) constitute a basis for H^{1}(f^{−1}(B), ℂ). Next, we will analyze the image of f^{∗}. It is clear that f^{∗}([dh_{i}/h_{i}]) = ∑_{j=1}^{n_{i}} α_{ij} [dg_{ij}/g_{ij}], hence these relations generate freely the kernel of r. Therefore, the set {r([dg_{ij}/g_{ij}])} (1 ≤ i ≤ k, 1 ≤ j ≤ n_{i} − 1) constitute a basis for H^{1}(F_{a_{0}}, ℂ)^{ρ}. This ends the proof in this first case.

(2.3) The general case. Assume that f = p ∘ g as in the introduction.

Let Λ_{f} be a hypersurface in ℂ^{n−1} in such a way that f is a locally trivial fibration over ℂ^{n−1}\Λ_{f}, g is a locally trivial fibration over Λ_{g} := p^{−1}(Λ_{f}), and the restriction p_{r} : ℂ^{n−1}\Λ_{g} → ℂ^{n−1}\Λ_{f} of p is a topological covering (of connected spaces). Set H := π_{1}(ℂ^{n−1} \ Λ_{g}, b_{0}) and G := π_{1}(ℂ^{n−1} \ Λ_{f}, a_{0}), where p(b_{0}) = a_{0} ∈ ℂ^{n−1}\Λ_{f}. Then one can consider the corresponding monodromy representations ρ_{G} : G → AutH^{1}(F_{a_{0}}, ℂ) and ρ_{H} : H → AutH^{1}(G_{b_{0}}, ℂ), where F_{a_{0}} := f^{−1}(a_{0}), G_{b_{0}} := g^{−1}(b_{0}). Moreover, p_{r} induces a monomorphism at the level of the fundamental groups, namely p_{∗} : H ↪ G. By the definition of the monodromy representations, and the definition of the induced representation (see, e.g. [FH], §3.3), one has the following connection between the two representations. (The proof is left to the reader.)

(2.4) Fact. ρ_{G} is the induced representation of ρ_{H} via p_{∗} : H ↪ G.

Let d be the topological degree of p. If [G : H] = d, then H^{1}(F_{a_{0}}, ℂ) can be identified with the direct sum ⊕^{d}H^{1}(G_{b_{0}}, ℂ) of d copies of H^{1}(G_{b_{0}}, ℂ). Moreover, by Frobenius Reciprocity (cf. [FH], page 35) one obtains that the invariant subspaces satisfies the next property: the map

H^{1}(G_{b_{0}}, ℂ)^{ρ_{H}} ∋ v ↦ (v, …, v) ∈ ⊕^{d}H^{1}(G_{b_{0}}, ℂ)

realizes an isomorphism H^{1}(G_{b_{0}}, ℂ)^{ρ_{H}} → H^{1}(F_{a_{0}}, ℂ)^{ρ_{G}}. This shows that the statement of the main theorem for g implies the result for f as well.

3. Applications

We consider two polynomail maps f : ℂ^{n} → ℂ^{n−1} and g : ℂ^{n} → ℂ. We will assume that F_{a} := f^{−1}(a) is connected for any a, and that the polynomial map (f, g) : ℂ^{n} → ℂ^{n} has a non-zero constant Jacobian. In particular, all the fibers F_{a} of f are smooth, hence (by the connectivity assumption) irreducible as well. Therefore, by our theorem, H^{1}(F_{a_{0}}, ℂ)^{ρ} = 0.

Corollary 3.1. Assume further that the fibers F_{a} are rational curves. Then (f, g) admits an inverse polynomial.

Proof. It is clear (cf. [BCW]) that it is enough to prove that (f, g) has degree one. Assume that the statement of the corollary is not true. Then for a generic a, the restriction g_{r} of g to F_{a} is not injective. Let \overline{F_{a}} be the smooth compactification of F_{a}, and fix an isomorphism t : \overline{F_{a}} → ℙ^{1} = ℂ ∪ {∞} in such a way that it takes no critical point or pole of g_{r} to {∞}. Assume that g_{r} has k critical points P_{1}, …, P_{l} with finite critical values, and l poles C_{1}, …, C_{k}. Because g_{r} is not injective, both k and l are positive. Let t_{i} := t(C_{i}) and τ_{i} := t(P_{i}). We prefer to think about these points as values in ℂ = ℙ^{1} \{∞}; and about t as the coordinate function on ℂ. Finally, consider the meromorphic 1-form

ω_{a} := t^{∗}(l ∑_{i=1}^{k} dt/(t − t_{i}) − k ∑_{i=1}^{l} dt/(t − τ_{i}))

on F_{a}, defined for any generic a. First notice that the Jacobian assumption about (f, g) guarantees that all the points {C_{i}}_{i} and {P_{i}}_{i} are situated in \overline{F_{a}} \ F_{a}. Therefore, ω_{a} is a holomorphic form on F_{a}. By the formula above, ω_{a} has k + l simple poles (and no other poles) with residues l at the points C_{1}, …, C_{k} and with residues −k at the points P_{1}, …, P_{l}. Notice that these properties completely determine a meromorphic form. In particular, ω_{a} is independent of the choice of t. Moreover, by the above residue properties, it is clear that it represents a non-trivial cohomology class of F_{a}. But, since it can be defined in a unique way for any generic a, this cohomology class is evidently monodromy invariant. But this leads to a contradiction. □

Corollary 3.2. Assume that (f, g) is a polynomial map as above. Then the generic fibers F_{a} cannot be (birationally equivalent to) an elliptic curves.

Proof. Again, assume that the statement is not true, i.e. the smooth compactification \overline{F_{a}} of the generic fiber is elliptic. Then there exist a non-zero holomorphic form ω̃_{a} on \overline{F_{a}} which is never vanishing. Such a form is uniquely determined up to a multiplicative constant. Let ω_{a} be its restriction to F_{a} and write ω_{a} = h_{a}dg for some function h_{a}. Because of the Jacobian condition, h_{a} does not have any pole or zero on F_{a}. Hence the 1-form θ_{a} = dh_{a}/h_{a} on F_{a} is well-defined. Moreover, its class is non-zero (indeed, h_{a} : F_{a} → ℂ^{∗} induces a monomorphism at cohomology level and dh_{a}/h_{a} = h_{a}^{∗}(dz/z)). Since in dh_{a}/h_{a} the ambiguity about the multiplicative constant disappears, it is determined in a unique way for any generic a. Hence it defines an invariant class, which leads again to a contradiction. □

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András Némethi The Ohio State University 231 W. 18th Avenue Columbus, OH 43210, USA.

and Alfréd Rényi Institute of Mathematics POB 127, H-1364 Budapest, Hungary nemethi@math.ohio-state.edu

István Sigray Eötvös University Budapest Department of Analysis H 1117 Budapest Pázmány P. Sétány 1/B Hungary sigray@cs.elte.hu