On the Monodromy at Infinity of a Polynomial Map

R. García López^{1} and A. Némethi^{2}

Universidad de Barcelona, Dept. de Algebra y Geometría. Gran Via, 585, 08007 Barcelona, Spain.

The Ohio State University, 231 West 18th Avenue, Columbus OH 43210, USA.

In this paper we study the semi-simple and unipotent part of the monodromy at infinity of a polynomial map which satisfies a natural restriction.

§1. Introduction.

(1.1) Let f : C^{n+1} → C be a map given by a polynomial with complex coefficients which will be also denoted by f. It is known (see, for example, [17, Appendix A1]) that there is a finite set Γ ⊂ C such that the map

f |_{C^{n+1}−f^{−1}(Γ)}: C^{n+1} − f^{−1}(Γ) → C − Γ

is a locally trivial C^{∞}-fibration. The smallest set Γ veryfing this condition is called the bifurcation set of f and will be denoted by Γ_{f}. Γ_{f} contains the set Σ_{f} of critical values of f but it might be bigger, since f is not a proper map. For example, if f = x(xy − 1) then Σ_{f} = ∅ but Γ_{f} = {0}.

Fix t_{0} ∈ C such that |t_{0}| > max{|γ| : γ ∈ Γ_{f}}. The geometric monodromy associated with the path s ↦ t_{0}e^{2πis}, s ∈ [0, 1] is a diffeomorphism of f^{−1}(t_{0}) onto itself which induces an isomorphism:

T_{f,Z}^{∞} : H^{n}(f^{−1}(t_{0}), Z) → H^{n}(f^{−1}(t_{0}), Z)

that will be called the monodromy of f at infinity. It follows from the monodromy theorem (e.g. in the form it is stated in [4, III.2.3]) that the eigenvalues of T_{f}^{∞} := T_{f,Z}^{∞} ⊗ Id_{C} are roots of unity.

1Supported by the EEC within the framework of the Human Capital and Mobility Program and by DGICYT, PB 91-0231-C02-01

2Partially supported by the Netherlands Organisation for the Advancement of Scientific Research N.W.O.

T_{f,Z}^{∞} is an invariant of the right equivalence class of f (we say that f, g : C^{n+1} → C are right equivalent if there is a diffeomorphism Φ : C^{n+1} → C^{n+1} such that f = g∘Φ), in particular it is also an invariant of the embedded affine variety {f = 0} ⊆ C^{n+1}. One expects that the study of T_{f}^{∞} can be useful in order to have a better understanding of the classification of polynomial maps up to right equivalence.

In this paper we study the monodromy at infinity of polynomials f ∈ C[X_{1}, …, X_{n+1}] verifying the following condition:

(∗) For t ∈ C − Σ_{f}, the closure in P^{n+1} of the affine hypersurface {f = t} ⊆ C^{n+1} is non-singular.

For the sake of brevity, a polynomial f verifying condition (∗) will be called a (∗)-polynomial.

In §2 we list the main properties of (∗)-polynomials. The most important ones are the following: the fibers of the map f : C^{n+1} → C have the homotopy type of a bouquet of n-spheres, and the fibration provided by f over a circe of large radius in C is equivalent to a fibration of type S^{2n+1} − K → S^{1} (i.e. to an open book decomposition). We also prove that T_{f}^{∞} is determined by the highest degree form f_{d} of f. If X_{f}^{∞} ⊆ P^{n} denotes the hypersurface given by f_{d} = 0, it is easy to see that X_{f}^{∞} has only isolated singularities. The main goal of this paper is to obtain information about the complex monodromy of f at infinity in terms of topological invariants of the embedded hypersurface X_{f}^{∞} ⊆ P^{n}.

In §3 we study the semi-simple part of T_{f}^{∞} (equivalently, its characteristic polynomial char_{f}^{∞}), and we prove that it is completely determined by local data: it depends only on the number of variables, the degree of f, and the characteristic polynomials of the (local) monodromies of the singularities of X_{f}^{∞}.

In contrast to this, we show in §4 that the unipotent part of T_{f}^{∞} does not depend only on local data, but also on the position of the singularities of X_{f}^{∞} in P^{n}. For example, it is proved in (4.6) that the number of Jordan blocks of T_{f}^{∞} associated with the eigenvalue 1 is the (n − 1)-th Betti number b_{n−1}(X_{f}^{∞}) if n is even and b_{n−1}(X_{f}^{∞}) − 1 if n is odd. On the other hand, it is well-known that the middle Betti number of a hypersurface with isolated singularities is not a purely local invariant but it depends on the position of the singular points (see for example [26], [5]). A concrete example showing how this reflects on the monodromy at infinity is given in (6.4).

In general, it is not possible to determine T_{f}^{∞} in terms of the Betti numbers of X_{f}^{∞} and local data, because even when the position of the singularities of X_{f}^{∞} does not have any influence on its Betti numbers (for example if X_{f}^{∞} is an irreducible curve), it can have an influence on the block structure of the monodromy at infinity. In §6 we exemplify the case when X_{f}^{∞} is a plane sextic with six cusps (Zariski’s example). It is well known that the position of the cusps determines the Betti numbers of the 6-fold cyclic covering of P^{2} ramified along X_{f}^{∞} (and then it determines whether the fundamental group of P^{2} − X_{f}^{∞} is abelian or not). In (6.5) we compute the unipotent part of T_{f}^{∞} in terms of the Betti numbers of the 6-fold cyclic covering.

We study the nilpotent (or unipotent) part of T_{f}^{∞} in two steps. In §4 we construct a compactification of the fibration of f at infinity, and we prove that its monodromy T determines completely T_{f}^{∞}. In §5 we consider a projective map π′ : X′ → D which is a fibration over the punctured disc D − {0} with monodromy T^{−d}. Moreover, the singularities of X′ and (π′)^{−1}(0) are isolated. This allows us to compute the transformation T^{d}: while (T^{d})_{≠1} is determined in terms of the local monodromies of the singularities of X_{f}^{∞}, the block structure of (T^{d})_{1} is given by the weight filtration (of the mixed Hodge structure) of the d-fold cyclic covering of P^{n} branched along X_{f}^{∞}.

By the correspondance proved in §4, the results of §5 give much information about the unipotent part of T_{f}^{∞}.

In the computation of (T^{d})_{1} the results of the Appendix (written by the first author and J.H.M. Steenbrink) are crucial.

(1.2) Unless otherwise stated, all cohomology and homology groups have coefficients in the field C of complex numbers. The following notations will be used through the paper:

• B_{R} = {(x_{1}, …, x_{n+1}) ∈ C^{n+1} | ∑ |x_{i}^{2}| < R}, B̄_{R} its closure, ∂B̄_{R} its boundary.

• D_{r} = {t ∈ C | |t| < r}, D̄_{r} its closure, S_{r} = ∂D̄_{r}.

• Given f ∈ C[X_{1}, …, X_{n+1}] the gradient of f will be denoted ∂f = (∂f/∂x_{1}, …, ∂f/∂x_{n+1}). If C^{n+1} ↪ P^{n+1} is the embedding given by (x_{1}, …, x_{n+1}) ↦ [1 : x_{1} : …: x_{n+1}] the hyperplane {x_{0} = 0} ⊂ P^{n+1} will be denoted by H^{∞}. If f_{d} is the highest degree form of f we will denote the hypersurface in H^{∞} given by f_{d} = 0 by X_{f}^{∞} or by X^{∞} if it is clear from the context which is the polynomial we are referring to. We always assume that d > 1.

• Let H be a finitely dimensional C-vector space, φ : H → H a linear map, λ ∈ C a complex number. We denote by H_{λ} the space of generalized eigenvectors of eigenvalue λ, i.e.

H_{λ} = {x ∈ H | ∃n ∈ N^{∗} with (φ − λ · Id)^{n}x = 0}

and φ_{λ} := φ_{|H_{λ}} : H_{λ} → H_{λ}. We denote by #_{k}φ_{λ} the number of k-dimensional Jordan blocks of φ_{λ}, #φ_{λ} = ∑_{k≥1} #_{k}φ_{λ} and #_{k}φ = ∑_{λ∈C} #_{k}φ_{λ}.

• If l > 0 is an integer, we denote by c_{l}(φ) : H^{⊕l} → H^{⊕l} the linear map defined by c_{l}(φ)(x_{1}, …, x_{l}) = (φ(x_{l}), x_{1}, …, x_{l−1}).

§2. Polynomials with good behaviour at infinity.

(2.1) In this section we will list the main properties of the (∗)-polynomials and we will prove that the highest degree form of a (∗)-polynomial determines its behaviour at infinity. For technical reasons, it will be more convenient to reformulate the (∗) condition as follows: Let f ∈ C[X_{1}, …, X_{n+1}] be a polynomial of degree d and denote f = f_{d} + f_{d−1} + … its decomposition into homogeneous components. Then f is a (∗)-polynomial if and only if

{x ∈ C^{n+1} | ∂f_{d}(x) = f_{d−1}(x) = 0 } = {0}.

The proof is easy and it is left to the reader. The (∗)-polynomials have a number of good properties which are summarized in the following:

(2.2) Theorem ([13], [15]): Let f be a (∗)-polynomial. Then:

i) The bifurcation set Γ_{f} is exactly the set Σ_{f} of critical values.

ii) The singular fibers f^{−1}(s) (s ∈ Σ_{f}) have only isolated singularities.

For s ∈ Σ_{f}, denote by μ_{s} the sum of the Milnor numbers of the isolated singularities of f^{−1}(s). Set μ^{∞} = Σ_{s∈Σ_{f}} μ_{s}. iii) Any fiber f^{−1}(s) has the homotopy type of a bouquet of n-dimensional

spheres. The number of spheres in the generic fiber is μ^{∞}, the number of spheres in a singular fiber f^{−1}(s) is μ^{∞} − μ_{s}.

iv) For any r > 0 with the property that Σ_{f} ⊂ D_{r}, there exists R_{0} ≫ 0 such that for any t ∈ D_{r}, R ≥ R_{0}, f^{−1}(t) intersects ∂B̄_{R} transversely and the restriction

f : (f^{−1}(S_{r}) ∩ B̄_{R}, f^{−1}(S_{r}) ∩ ∂B̄_{R}) → S_{r}

is a C^{∞}-locally trivial fibration of pairs of spaces.

The fibration f : f^{−1}(S_{r}) ∩ B̄_{R} → S_{r} is equivalent to the fibration f : f^{−1}(S_{r}) → S_{r} and it will be called the fibration of f at infinity. The fibration f : f^{−1}(S_{r}) ∩ ∂B̄_{R} → S_{r} extends to a C^{∞}-trivial fibration f : f^{−1}(D̄_{r}) ∩ ∂B̄_{R} → D̄_{r}.

v) There exists R′_{0} ≫ 0 such that for any R′ ≥ R′_{0}

φ = \frac{f}{∥f∥} : ∂B̄_{R′} − f^{−1}(0) → S^{1}

is a C^{∞}-locally trivial fibration (called the Milnor fibration at infinity), which is equivalent to the fibration of f at infinity.

vi) Let X_{f}^{∞} be the intersection of the hyperplane at infinity H^{∞} with the projective closure X_{t} = \overline{f^{−1}(t)} ⊆ P^{n+1} of any fiber f^{−1}(t). Then the hypersurface X_{f}^{∞} has only isolated singularities.

Proof: In [6] it is proved that a (∗)-polynomial (in the sense of (2.1)) is quasitame, a condition introduced by the second author in [13]. Now i)-iv) follow from [13], v) follows from [15] and vi) from an easy verification. □

(2.3) Let (F, ∂F) denote the fiber of the fibration (2.2.iv). Then the last part of iv) implies that there is a smooth representative T_{geom} of the geometric monodromy which is the identity on ∂F. This allows us to define a variation map V ar : H^{n}(F) → H_{c}^{n}(F) by V ar[ω] = [T_{geom}^{∗}(ω) − ω]. (For the definition at the level of integer homology, see for example [1]). The variation map fits in the following diagram:

[figure: H_{c}^{n}(F); H_{c}^{n}(F); H^{n}(F); H^{n}(F); V ar; k; k; T_{f,c}^{∞} − Id; T_{f}^{∞} − Id]

where T_{f}^{∞} and T_{f,c}^{∞} are the corresponding monodromies and k is the natural map. The following result will be crucial in the study of the unipotent part of T_{f}^{∞}:

(2.4) Proposition: V ar is an isomorphism.

Proof: Since the fibration at infinity is equivalent to the Milnor fibration at infinity (2.2.v), V ar is the variation map of a fibration of type S^{2n+1}−K → S^{1}. Similarly as in the local case of isolated hypersurface singularities, the variation map is an isomorphism by Alexander duality (see [1] for more details). Actually, (2.4) is equivalent to the non-degeneracy of the Seifert form of the open book decomposition. □ (2.5) Remark: Theorem (2.2) and proposition (2.4) hold not only for (∗)- polynomials but for a larger class of polynomials which includes the ”tame” ([2]) and ”quasi-tame” polynomials ([13]), the ”M-tame” polynomials ([15]) and the convenient polynomials, non-degenerate with respect to their Newton boundary at infinity ([8]).

(2.6) Theorem: Let f = f_{d} + f_{d−1} + ... , g = g_{d} + g_{d−1} + … be two (∗)- polynomials of degree d such that f_{d} = g_{d}. Then the fibrations at infinity of f and g are equivalent (in the sense of [25], p. 11). In particular all the invariants, introduced in (2.2) and (2.3) for f and g, are equivalent.

Proof: Notice that the set of (∗)-polynomials with fixed highest degree form f_{d} form a connected, smooth, quasiprojective variety. Thus, in order to prove the theorem, it is enough to prove the following claim:

Claim: Let f = f_{d} +f_{d−1} +… be a (∗)-polynomial. Fix 0 ≤ i ≤ d−1 and consider the family of polynomials f_{s} = f + (s − 1)f_{i}, s ∈ C. Fix 1 ≫ η > 0 such that each f_{s} is a (∗)-polynomial for |s − 1| ≤ η. Then there exist r ≫ 0 and R_{0} ≫ 0 such that:

i) Σ_{f_{s}} ⊆ D_{r} for |s − 1| ≤ η;

ii) f_{s}^{−1}(t) intersects ∂B̄_{R} transversely for |s − 1| ≤ η, R ≥ R_{0} and t ∈ D̄_{r}.

Now, the proof of the theorem is the following: Set

E = {(x, t, s) ∈ B̄_{R_{0}} × S_{r} × B_{1}(η) | f_{s}(x) = t},
∂E = {(x, t, s) ∈ E | x ∈ ∂B̄_{R_{0}}},

where B_{1}(η) = {s | |s−1| < η}. Then the projection q : (E, ∂E) → S_{r}×B_{1}(η) is a locally trivial fibration and for any s ∈ B_{1}(η), q_{|q^{−1}(S_{r}×{s})} is the fibration of f_{s} at infinity. Then the result follows from [25, p. 53].

Proof of the claim: Assume that i) is not true. Then, by the curve selection lemma ([11], [15]), there exist real analytic curves x(t) ∈ C^{n+1}, and s(t) ∈ C (0 < t < ε) such that |s(t) − 1| ≤ η, ∂f_{s(t)}(x(t)) ≡ 0, and lim_{t→0} f_{s(t)}(x(t)) = ∞. The last limit implies that lim_{t→0} ∥x(t)∥ = ∞. Put x(t) = t^{−n}y(t) with n > 0, y(t) = y_{0} + ty_{1} + …, y_{0} ≠ 0. Then:

(1) ∂f_{d}(y(t)) + t^{n}∂f_{d−1}(y(t)) + … + s(t)t^{(d−i)n}∂f_{i}(y(t)) + … ≡ 0.

In particular ∂f_{d}(y_{0}) = 0. Identity (1) gives

(2) ∂f_{d}(y(t)) + t^{n} · c · ∂f_{d−1}(y_{0}) ≡ 0 (mod t^{n+1}),

where c = 1 if i < d − 1 and c = s(0) if i = d − 1. This identity, multiplied by y(t), rewritten using the Euler–relations, differentiated with respect to t, and compared with its initial form (2), gives:

n · t^{n−1} · c · (d − 1)f_{d−1}(y_{0}) ≡ 0 (mod t^{n}).

Therefore f_{d−1}(y_{0}) = 0, which contradicts condition (∗).

Part ii) follows from a similar argument. Fix an r ≫ 0 which satisfies i). Assume that ii) is not true for η, r and any R_{0} ≫ 0. Then there exist analytic curves x(t) ∈ C^{n+1}, λ(t) ∈ C and s(t) ∈ C, (0 < t < ε), with |s(t) − 1| ≤ η, |f_{s(t)}(x(t))| ≤ r, lim_{t→0} ∥x(t)∥ = ∞, and

(3) ∂f_{s(t)} · (x(t)) = λ(t) · \overline{x(t)}.

Let x(t) = t^{−n}y(t) as above. Since f_{s(t)}(x(t)) has order 0,

∂f_{s(t)}(x(t)) · x′(t) + s′(t) · f_{i}(x(t)) ≡ 0 (mod t^{0}).

Hence λ(t) · \overline{x(t)} · x′(t) + s′(t) · t^{−ni}f_{i}(y(t)) ≡ 0 (mod t^{0}), which gives λ(t) ≡ 0 (mod t^{(2−i)n+1}). Now (3) gives (mod t^{n+1}) the same equation as (2) which gives again the contradiction ∂f_{d}(y_{0}) = f_{d−1}(y_{0}) = 0. □

(2.7) Theorem (2.6) says that the behaviour of f = f_{d} + f_{d−1} + … at infinity depends only on f_{d}, in particular f can be replaced by any polynomial f′ = f_{d} + f′_{d−1} where f′_{d−1} is a polynomial such that its zero set in P^{n} does not intersect Sing(X^{∞}). For example, we can take f′_{d−1} = l^{d−1} where l is a generic linear form. This fact gives the hope that the topological invariants of f associated to its behaviour at infinity can be explicitly described in terms of invariants of the embedded hypersurface X^{∞} ⊆ P^{n}.

§3. The semi-simple part of T_{f}^{∞}.

(3.1) Let f = f_{d} + f_{d−1} be a (∗)-polynomial. We introduce some notations. Let Sing(X^{∞}) = {p_{1}, …, p_{k}}. For 1 ≤ i ≤ k, let g_{i} : (H^{∞}, p_{i}) → (C, 0) be a local equation defining the isolated hypersurface singularity germ (X^{∞}, p_{i}). We denote by μ_{i} its local Milnor number, by F_{i} its local Milnor fiber, and by T_{i} its algebraic monodromy acting on H^{n−1}(F_{i}). In this section we determine the characteristic polynomial of T_{f}^{∞} (or equivalently, its semi-simple part) in terms of the characteristic polynomials of the local algebraic monodromies {T_{i}}_{i=1,...,k}. (3.2) First assume that X^{∞} is non-singular. Then f_{d} : (C^{n+1}, 0) → (C, 0) defines an isolated singularity and, by similar argument as in (2.6), the fibration of f at infinity is the same as the fibration of f_{d} at infinity, which is identical with the local Milnor fibration of the germ f_{d} defined at 0. In particular, its Milnor number is μ_{gen}^{∞} = (d−1)^{n+1}, its monodromy has (finite) order d, and its characteristic polynomial is

char_{gen}(λ) := det(λ · Id − T_{gen}^{∞}) = (λ − 1)^{(−1)^{n+1}}(λ^{d} − 1)^{\frac{(d−1)^{n+1}−(−1)^{n+1}}{d}} .

Set char_{f}^{∞}(λ) = det(λ · Id − T_{f}^{∞}), char_{i}(λ) = det(λ · Id − T_{i}). Then we have:

(3.3) Theorem: Assume that f is a (∗)-polynomial. Then, with the notations introduced above one has:

char_{f}^{∞}(λ) = char_{gen}(λ) · ∏_{i=1}^{k} \frac{char_{i}(λ^{d−1})}{(λ^{d} − 1)^{μ_{i}}}.

In particular, μ^{∞} + ∑_{i=1}^{k} μ_{i} = μ_{gen}^{∞}.

Before we start the proof of theorem (3.3), we make some preparations:

(3.4) Lemma: Consider the family of polynomials f_{s} = f_{d} + sf_{d−1}, where s ∈ C. There exists r > 0 such that the set of critical values of f_{s} is contained in D_{r} for any s with |s| ≤ 1.

The proof is similar to the proof of the Claim in (2.6), and it is left to the reader.

(3.5) We recall the definition of the zeta function of a locally trivial fibration E → S^{1} over the one-dimensional circle S^{1}. Let F be its fiber and T^{q} : H^{q}(F) → H^{q}(F) the algebraic monodromies induced by its characteristic map. Then define:

ζ(E → S^{1}) = ∏_{q} det(λ · Id − T^{q})^{(−1)^{q}}.

If f is an arbitrary polynomial and r is large enough so that Γ_{f} ⊂ D_{r}, then ζ^{∞}(f) is, by definition, ζ(f^{−1}(S_{r}) → S_{r}).

(3.6) Proof of (3.3): Fix r big enough so that the conclusion of Lemma 3.4 holds for r. For any s ∈ C with |s| ≤ 1 and t ∈ S_{r}, set

X_{t,s} = {[x] ∈ P^{n+1} | f_{d}(x_{1}, …, x_{n+1}) + sx_{0}f_{d−1}(x_{1}, … x_{n+1}) = tx_{0}^{d}}.

X_{t,s} is the projective closure of the affine variety f_{s}^{−1}(t). Since for any s ≠ 0 the polynomial f_{s} satisfies (∗), for any s ≠ 0 the space X_{t,s} is non-singular. The intersection X_{t,s} ∩ H^{∞} is exactly X^{∞}, in particular it has only isolated singularities (cf. 2.2.vi). On the other hand, X_{t,0} is singular with isolated singularities exactly at the singular points of X^{∞}.

Let B_{i} be a small open “ball” in P^{n+1} with center at p_{i}, (1 ≤ i ≤ k). More precisely, consider a real analytic function r_{i} defined in a neighborhood of p_{i} with non-negative values such that r_{i}^{−1}(0) = {p_{i}}, and take B_{i} = r_{i}^{−1}([0, η_{0})). Here η_{0} is small enough such that B_{i}’s are disjoint and the following conditions hold:

i) r_{i}^{−1}(η) is smooth and intersects X_{t,0}, H^{∞} and X^{∞} transversely for any η ≤ η_{0}, t ∈ S_{r} and i = 1, …, k,

ii) The “ball” r_{i}^{−1}([0, η]) and its intersections with X_{t,0}, H^{∞} and X^{∞} retract to p_{i} for any η ≤ η_{0}, (in other words, B_{i} is a “Milnor ball” at p_{i} for the analytic sets X_{t,0}, H^{∞} and X^{∞}). Fix η_{0} and the balls {B_{i}}_{i=1,...,k}. Let B̄_{i} be the closure of B_{i}, and ∂B̄_{i} its boundary.

Now we are going to consider balls B̄_{R} in the affine space C^{n+1}. Choose R_{0} big enough so that one has:

iii) ∂B̄_{R} intersects f_{0}^{−1}(t) transversely for any R ≥ R_{0} and t ∈ S_{r},

iv) ∂B̄_{R} intersects ∂B̄_{i} transversely for any R ≥ R_{0} and i = 1, …, k.

Fix a R_{0} with these properties and set C_{R_{0}} = C^{n+1} −B_{R_{0}}. Since f_{s}^{−1}(t) is smooth for any s, t and f_{0}^{−1}(t) intersects ∂B̄_{R_{0}} transversely, there exists 1 ≫ ε > 0 such that for any s ∈ C with |s| ≤ ε one has:

v) f_{s}^{−1}(t) intersects ∂B̄_{R_{0}} transversely for any t,

vi) X_{t,s} intersects ∂B̄_{i} transversely for any t and i = 1, …, k.

Obviously, in general R_{0} does not satisfy the condition (2.2.iv) for all s.

With these choices, the map:

(f_{s}^{−1}(S_{r}), f_{s}^{−1}(S_{r}) ∩ B̄_{R_{0}}, f_{s}^{−1}(S_{r}) ∩ C_{R_{0}}) → S_{r}, |s| ≤ ε

is a locally trivial fibration of a triple of spaces. By the transversality conditions one has the equivalence of the fibrations:

f_{0}^{−1}(S_{r}) ∩ B_{R_{0}} → S_{r} and f_{ε}^{−1}(S_{r}) ∩ B_{R_{0}} → S_{r}

respectively of

(f_{0}^{−1}(S_{r}) ∩ ∂B̄_{R_{0}} → S_{r}) and (f_{ε}^{−1}(S_{r}) ∩ ∂B̄_{R_{0}} → S_{r}).

By a Mayer-Vietoris argument one has:

(5) \frac{ζ^{∞}(f_{ε})}{ζ^{∞}(f_{0})} = \frac{ζ(f_{ε}^{−1}(S_{r}) ∩ C_{R_{0}} → S_{r})}{ζ(f_{0}^{−1}(S_{r}) ∩ C_{R_{0}} → S_{r})}.

Set:

F = {(x, t, s) ∈ C_{R_{0}} × S_{r} × D̄_{ε} : f_{s}(x) = t}.

If A ⊂ C_{R_{0}} and s ∈ D̄_{ε} then set F(A)_{s} := F ∩ (A × S_{r} × {s}). Denote:

Y_{s} = F((∪_{i=1}^{k}B̄_{i})∩C_{R_{0}})_{s}, ∂Y_{s} = F((∪_{i=1}^{k}∂B̄_{i})∩C_{R_{0}})_{s}, Z_{s} = F(C_{R_{0}}−∪_{i=1}^{k}B_{i})_{s}.

Consider the map F → S_{r} × D̄_{ε} induced by the projection. For any s ∈ D̄_{ε} the induced map (X_{s}, Y_{s}, Z_{s}) → S_{r} is a locally trivial fibration of a triple of spaces, and the fibrations Z_{s} → S_{r} and ∂Y_{s} → S_{r} are independent on s. Then, by Mayer-Vietoris one has:

(6) ζ(F(C_{R_{0}})_{s} → S_{r}) = \frac{ζ(Y_{s} → S_{r}) · ζ(Z_{s} → S_{r})}{ζ(∂Y_{s} → S_{r})}.

for any |s| ≤ ε. Now (5) and (6) give

(7) \frac{ζ^{∞}(f_{ε})}{ζ^{∞}(f_{0})} = \frac{ζ(F(C_{R_{0}})_{ε} → S_{r})}{ζ(F(C_{R_{0}})_{0} → S_{r})} = \frac{ζ(Y_{ε} → S_{r})}{ζ(Y_{0} → S_{r})}.

Notice that the right hand side of (7) is completely local, it is concentrated in the balls {B_{i}}_{i}. The rest of the proof is now devoted to compute ζ(Y_{0} → S_{r}) and ζ(Y_{ε} → S_{r}). Since the balls {B_{i}}_{i} are disjoint, these zeta functions are products over the singular points p_{i} of X^{∞}: ζ(Y_{s} → S_{r}) = ∏_{i} ζ(Y_{si} → S_{r}), where Y_{si} = Y_{s} ∩ B_{i} and s = 0 or = ε.

Fix a point p_{i} ∈ Sing(X^{∞}) and choose coordinates so that p_{i} = [0 : …: 0 : 1] and f_{d−1} = x_{n+1}^{d−1} (cf. 2.6-2.7). We recall that we fixed a circle of big radius S_{r} and after that we fixed the balls B_{i} and R_{0} (thus R_{0} and B_{i} depend on the choice of r). In local coordinates (y_{0}, …, y_{n}) (using the notation y_{i} = x_{i}/x_{n+1} for i = 0, …, n and y = (y_{1}, …, y_{n})) one has:

C_{R_{0}} = {(y_{0}, y) | 1 + ∥y∥^{2} ≥ R_{0}|y_{0}|^{2}, y_{0} ≠ 0},

and we can assume that B̄_{i} = {|y_{0}|^{2}+∥y∥^{2} ≤ ρ_{i}} for some small ρ_{i} > 0. Then Y_{0i} = {(y_{0}, y, t) ∈ (B̄_{i} ∩ C_{R_{0}}) × S_{r} | g_{i}(y) = ty_{0}^{d}} and the map Y_{0i} → S_{r} is induced by the first projection. We will show that ζ(Y_{0i} → S_{r}) = 1. For this consider the neighborhood N_{i} = {(y_{0}, y) | 1 ≥ R_{0}|y_{0}|^{2} , |y_{0}|^{2} + ∥y∥^{2} ≤ ρ_{i}}. Then it is not difficult to see that

Y′_{0i} := {(y_{0}, y, t) ∈ N_{i} × S_{r} | y_{0} ≠ 0 , g_{i}(y) = ty_{0}^{d}} → S_{r}

is a subbundle of Y_{0i} → S_{r} which is an equivariant strong deformation retract. In particular, ζ(Y_{0i} → S_{r}) = ζ(Y′_{0i} → S_{r}).

Now set ρ = \sqrt{1/R_{0}} and consider the map Y′_{0i} → S_{r} × D_{ρ}^{∗} given by (y_{0}, y, t) → (t, y_{0}). By construction, this is a fiber bundle (with fiber F_{i}), thus ζ(Y′_{0i} → S_{r}) = 1 by [14, 3.3.9].

The next step is the computation of ζ(Y_{εi} → S_{r}). Let r, R_{0}, ρ_{i} be as above. Then ε is fixed and is sufficiently small with respect to r, R_{0}, ρ_{i} (see conditions v) and vi) above). Then:

Y_{εi} = {(y_{0}, y, t) ∈ (B̄_{i} × C_{R_{0}}) × S_{r} | y_{0} ≠ 0 and ty_{0}^{d} − εy_{0} = g_{i}(y)}.

Consider, similarly as above,

Y′_{εi} = {(y_{0}, y, t) ∈ N_{i} × S_{r} | y_{0} ≠ 0 and ty_{0}^{d} − εy_{0} = g_{i}(y)}

with the projection onto S_{r}. Then Y_{εi} → S_{r} and Y′_{εi} → S_{r} are equivalent fiber bundles. Now take the projection τ_{i} : Y′_{εi} → S_{r} × D_{ρ}^{∗} given by (y_{0}, y, t) → (t, y_{0}). This is a fiber bundle over S_{r} × D_{ρ}^{∗} − {(t, y_{0}) | ty_{0}^{d} = εy_{0}} with fiber F_{i} and the fiber of τ_{i} over any point of Δ := {(t, y_{0}) ∈ S_{r} × D_{ρ}^{∗} | ty_{0}^{d} = εy_{0}} is contractible. For a fixed value t = t_{0} ∈ S_{r}, the punctured disk {t_{0}} × D_{ρ}^{∗} intersects Δ in (d − 1) points q_{1}, …, q_{d−1}. If t_{0} ∈ R_{+} (i.e. if t_{0} = r), these points are:

q_{k} = (t_{0} , e^{2πik/d−1} · ρ_{ε}) k = 1, …, d − 1.

They are situated on the circle

S(t_{0}) = {(t, y_{0}) ∈ S_{r} × D_{ρ}^{∗} | t = t_{0} , |y_{0}| = ρ_{ε}}.

Consider also the points r_{k} ∈ S(t_{0}), k = 1, …, d − 1:

r_{k} = (t_{0} , e^{2πi(k+ \frac{1}{2} )/d−1} · ρ_{ε}).

Now the fiber F_{εi} of Y′_{εi} over t_{0} is τ^{−1}({t_{0}}×D_{ρ}^{∗}). S(t_{0}) is a strong deformation retract of {t_{0}} × D_{ρ}^{∗} and the retraction can be lifted, therefore F_{εi} has the homotopy type of τ^{−1}(S(t_{0})). The points r_{k} lay on the arc q̂_{k}q_{k+1}, τ^{−1}(r_{k}) ≃ F_{i} and τ^{−1}(q_{k}) is contractible. Therefore τ^{−1}(q̂_{k}q_{k+1}) has the homotopy type of the suspension S(F_{i}) of F_{i}. In particular τ^{−1}(S(t_{0})) ≃ S^{1} ∨ (⋁_{d−1} S(F_{i})).

If we now lift the path given by α ↦ t_{0}e^{2πiα} ∈ S_{r}, α ∈ [0, 1], then the points q_{k}, respectively r_{k}, move on the path:

q_{k}(α) = (t_{0}e^{2πiα}, e^{2πi(−α+k)/(d−1)} · ρ_{ε}),

respectively on

r_{k}(α) = (t_{0}e^{2πiα}, e^{2πi(−α+k+ \frac{1}{2} )/(d−1)} · ρ_{ε}), α ∈ [0, 1].

Obviously r_{k}(0) = r_{k} and r_{k}(1) = r_{k−1} (with the notation r_{0} = r_{d−1}). Thus via the paths r_{2}(α), …, r_{d−1}(α), we can identify the fibers of the bundle τ over the points {r_{k}}_{k=1}^{d−1}. We determine now the geometric monodromy acting on S^{1} ∨(⋁_{d−1} S(F_{i})). It is clear that the action on S^{1} is trivial. By the above identifications, the action on ⋁_{d−1} S(F_{i}) is given by c_{d−1}(S(T_{geom})) (see (1.2)), where S(T_{geom}) is the suspension of the geometric monodromy T_{geom} induced by the loop γ = r_{1}(α) ∘ … ∘ r_{d−1}(α). If γ is given by α ↦ (t(α), y_{0}(α)) then the loops γ_{u} defined by α ↦ (t(α), uy_{0}(α)) for u ∈ [c, 1], (1 > c > 0) are homotopic to γ. Moreover, the image of the loop γ_{u} is on the torus T_{u} = S_{r} ×{y_{0} | |y_{0}| = uρ_{ε}} and for c ≤ u < 1 the torus T_{u} does not intersect Δ. Now, the fibration induced by τ over T_{c} is a pullback of a representative G_{i} : B_{i} → D_{ρ} of g_{i} via the map ξ : T_{c} → D_{ρ}^{∗} given by (t, y_{0}) ↦ −εy_{0} + ty_{0}^{d}.

Let L and M be oriented loops on T_{c} which generate its first homology group, i.e. L = [s ↦ (t_{0}, cρ_{ε}e^{2πis})] and M = [s ↦ (t_{0}e^{2πis}, cρ_{ε})]. Then we have [γ_{c}] = −L + (d − 1)M in H_{1}(T_{c}, Z). Now, first notice that the monodromy induced by M is trivial. To see this, notice that for c sufficiently small and y_{0} ≠ 0 fixed, the winding number of the loop α ↦ y_{0}(−ε + t_{0}e^{2πiα}y_{0}^{d−1}) (with respect to the origin) is zero. Therefore ξ_{∗}[M] is trivial in H_{1}(D_{ρ}^{∗}). On the other hand, ξ_{∗}[L] = 1 ∈ H_{1}(D_{ρ}^{∗}). Therefore T_{geom} is the inverse the geometric monodromy of g_{i}. The corresponding algebraic monodromy acting on H_{n}(⋁_{d−1} S(F_{i})) is A_{i} = c_{d−1}(T_{i}^{−1}). Thus ζ(Y_{εi} → S_{r}) = (det (λ · Id − A_{i}))^{(−1)^{n}} and since T_{i} is a real operator with eigenvalues on the unit circle det(λ − A_{i}) = char_{i}(λ^{d−1}). Then (7) reads:

(8) ζ^{∞}(f_{ε}) = ζ^{∞}(f_{0}) · ∏_{i=1}^{k} char_{i}(λ^{d−1})^{(−1)^{n}}.

Now, for s = 0, f_{s} is exactly the homogeneous polynomial f_{d}, considered as a germ f_{d} : (C^{n+1}, 0) → (C, 0). f_{d} defines a singularity with one-dimensional critical locus and its zeta function is computed in [21]:

(9) ζ(f_{d}) = ζ(f_{gen}) · ∏_{i=1}^{k}(λ^{d} − 1)^{μ_{i}(−1)^{n+1}},

where ζ(f_{gen}) is the zeta function of a generic homogeneous singularity of degree d, i.e. ζ(f_{gen}) = (1 − λ)(char_{gen}(λ))^{(−1)^{n}}. Since ζ^{∞}(f_{ε}) = (1 − λ) · (char_{f}^{∞})^{(−1)^{n}}, the result follows from (8) and (9). □

§4. The algebraic monodromy T_{f}^{∞} via a compactification.

(4.1) In the next two sections we will study the structure of the Jordan blocks of T_{f}^{∞} for a (∗)-polynomial f. For this purpose we will consider a fibration π : X → S_{r} which compactifies the fibration of f at infinity. The main result of this section is that the algebraic monodromy T_{f}^{∞} can be completely determined from the algebraic monodromy T of the projective fibration π (and conversely).

One interesting byproduct of this correspondence is that the number of Jordan blocks of T_{f}^{∞} corresponding to eigenvalue one is the (n − 1)-th Betti number b_{n−1}(X^{∞}) if n is even and b_{n−1}(X^{∞}) − 1 if n is odd. In particular, the unipotent part of the monodromy at infinity depends not only on local data associated to the singularities of X^{∞} but also on their position.

In the next section we determine the d-th power of T. This gives much information about the unipotent part of T_{f}^{∞} via the correspondence of this section. (4.2) We introduce some notations. Let f ∈ C[X_{1}, …, X_{n+1}] be a (∗)- polynomial. By (2.6) we can assume that f has the form f = f_{d} + x_{n+1}^{d−1}, where f_{d} is homogeneous of degree d. Fix r such that Σ_{f} ⊂ D_{r}. Set:

X = {([x_{0}, …, x_{n+1}], t) ∈ P^{n+1} × S_{r} | f_{d}(x) + x_{0}x_{n+1}^{d−1} = tx_{0}^{d} }

and let π : X → S_{r} be the second projection map. Put:

\begin{aligned}X^{∞} × S_{r} & = & \{([x], t) ∈ X | x_{0} = 0\} , \\ X_{t} & = & p^{−1}(t) (t ∈ S_{r}) , \end{aligned}  \begin{aligned}X^{0} & = & X − X^{∞} × S_{r}, \\ X_{t}^{0} & = & X_{t} ∩ X^{0} (t ∈ S_{r}). \end{aligned}

Fix t_{0} ∈ S_{r}. Then π : (X , X^{0}) → S_{r} is a locally trivial fibration of pairs of spaces and π_{|X^{0}} is exactly the fibration of f at infinity. Let T : H^{n}(X_{t_{0}}) → H^{n}(X_{t_{0}}) be the algebraic monodromy of π. Property (2.2.v) implies that there exists a sufficiently large ball B_{R_{0}} and a representative T_{geom} : X_{t_{0}} → X_{t_{0}} of the geometric monodromy of π such that T_{geom |X_{t_{0}}−B_{R_{0}}} is the identity. Therefore, the diagram in (2.3) can be extended to the following diagram:

[figure: H^{n}(X_{t_{0}}); H^{n}(X_{t_{0}}^{0}); H^{n}(X_{t_{0}}); H_{c}^{n}(X_{t_{0}}^{0}); T − Id; V ar; i^{∗}; i_{∗}]

where i_{∗} and i^{∗} are the natural maps. Recall that i^{∗}i_{∗}V ar = k ∘ V ar = T_{f}^{∞} − Id. (4.3) Since f is a (∗)-polynomial, X_{t_{0}} is a smooth hypersurface in P^{n+1}. Let S be the polarization form of H^{n}(X_{t_{0}}), i.e. S(α, β) = ∫_{X_{t_{0}}} α ∧ β for α, β ∈ H^{n}(X_{t_{0}}). Then it is well-known that H^{q}(X_{t_{0}}) = H^{q}(P^{n+1}) if q ≠ n and H^{n}(X_{t_{0}}) decomposes in a direct sum H^{n}(P^{n+1}) ⊕ P^{n}(X_{t_{0}}), orthogonal with respect to S.

Since the hyperplane section at infinity X^{∞} has only isolated singularities, the primitive decomposition H_{q}(X^{∞}) = H_{q}(P^{n}) ⊕ P_{q}(X^{∞}) satisfies P_{q}(X^{∞}) = 0 if q ≠ n − 1, n. The numbers p_{q}(X^{∞}) = dim P_{q}(X^{∞}) (q = n − 1, n), are in general non-zero, and in general, they depend on the position of the singularities of X^{∞}.

On the other hand, the Euler characteristic of X^{∞} does not depend on it. One has:

p_{n−1}(X^{∞}) − p_{n}(X^{∞}) = \frac{d − 1}{d}[(d − 1)^{n} − (−1)^{n}] − ∑_{i=1}^{k} μ_{i}.

(4.4) Since X_{t_{0}}^{0} has the homotopy type of a bouquet of n-spheres (cf. 2.2.iii), the exact sequence of cohomology with supports has the following form:

0 → H_{X^{∞}}^{n}(X_{t_{0}}) →^{j_{∗}} H^{n}(X_{t_{0}}) →^{i^{∗}} H^{n}(X_{t_{0}}^{0}) → H_{X^{∞}}^{n+1}(X_{t_{0}}) → H^{n+1}(X_{t_{0}}) → 0.

The sequence is equivariant with respect to the monodromy action. This action on H_{X^{∞}}^{n}(X_{t_{0}}), H_{X^{∞}}^{n+1}(X_{t_{0}}) and H^{n+1}(X_{t_{0}}) is the identity. Recall also the duality isomorphism H_{X^{∞}}^{∗}(X_{t_{0}}) ≃ H_{2n−∗}(X^{∞}). In the sequel we will identify H_{X^{∞}}^{n}(X_{t_{0}}) with its image in H^{n}(X_{t_{0}}). (4.5) Lemma: There is an equivariant direct sum decomposition

H^{n}(X_{t_{0}}) ≃ H_{X^{∞}}^{n}(X_{t_{0}}) ⊕ (H_{X^{∞}}^{n}(X_{t_{0}}))^{⊥},

which is orthogonal with respect to the form S. In particular, the monodromy T decomposes as T = Id ⊕ T′. Proof: Consider the orthogonal decomposition H_{X^{∞}}^{n}(X_{t_{0}}) = H^{n}(P^{n+1}) ⊕ P_{X^{∞}}^{n}(X_{t_{0}}) and the corresponding decomposition of j_{∗}:

j_{∗} = Id ⊕ j′ : H^{n}(P^{n+1}) ⊕ P_{X^{∞}}^{n}(X_{t_{0}}) → H^{n}(P^{n+1}) ⊕ P^{n}(X_{t_{0}}).

Then P_{X^{∞}}^{n}(X_{t_{0}}) has a natural polarized Hodge structure and the inclusion j′ is a morphism of polarized Hodge structures (where on P^{n}(X_{t_{0}}) the polarization is induced by S). It follows that the restriction of S to P_{X^{∞}}^{n}(X_{t_{0}}) is non-degenerate hence the result follows. □

We have a perfect pairing Q : H_{c}^{n}(X_{t_{0}}^{0}) ⊗ H^{n}(X_{t_{0}}^{0}) → C given by α ⊗ β ↦ ∫_{X_{t_{0}}^{0}} α ∧ β and the morphisms i_{∗} and i^{∗} are adjoint with respect to Q. Thus im(i_{∗}) ≃ (ker (i^{∗}))^{⊥} = (H_{X^{∞}}^{n}(X_{t_{0}}))^{⊥}. Since Q is compatible with the monodromy action this isomorphism is equivariant.

The following theorem describes the monodromy T_{f}^{∞} in terms of T and the Betti numbers of X^{∞}.

(4.6) Theorem :

a) For any λ ≠ 1, (T_{f}^{∞})_{λ} = T_{λ}.

b) Assume λ = 1. Then:

i) #_{1}(T_{f}^{∞})_{1} = b_{n}(X^{∞}) + p_{n−1}(X^{∞}) − #T_{1},

ii) #_{2}(T_{f}^{∞})_{1} = #_{1}T_{1} − b_{n}(X^{∞}),

iii) #_{l+1}(T_{f}^{∞})_{1} = #_{l}T_{1} for l ≥ 2.

In particular #(T_{f}^{∞})_{1} = dim(ker T_{f}^{∞} − Id) = p_{n−1}(X^{∞}).

Proof: Part a) follows from (4.4). Let V_{1} denote the composed map

H^{n}(X_{t_{0}}^{0})_{1} ⟶^{V ar_{1}} H_{c}^{n}(X_{t_{0}}^{0})_{1} ⟶^{(i_{∗})_{1}} im(i_{∗})_{1}.

By (2.4) we have that V_{1} is onto. By (4.2), (4.4) and (4.5) one has the following commutative diagram:

[figure: 0; im(i_{∗})_{1}; H^{n}(X_{t_{0}}^{0})_{1}; P_{n−1}(X^{∞}); 0; 0; im(i_{∗})_{1}; H^{n}(X_{t_{0}}^{0})_{1}; P_{n−1}(X^{∞}); 0; T′_{1} − Id; (T_{f}^{∞})_{1} − Id; 0; V_{1}]

Now part b) follows from this diagram and (4.5).□ (4.7) Corollary: Assume 1 is not an eigenvalue of any of the local monodromies T_{i} (i = 1, …, k), (cf. 3.1). Then #_{l}(T_{f}^{∞})_{1} = 0 for l > 1 (i.e. (T_{f}^{∞})_{1} = Id).

Proof: From (3.3) and from the Euler–characteristic formula (4.3) we deduce that p_{n−1}(X^{∞}) − p_{n}(X^{∞}) = dim H^{n}(X_{t_{0}}^{0})_{1}. Then the exact sequence (4.4) gives that P^{n}(X_{t_{0}})_{1} = 0, or (i^{∗})_{1} = (i_{∗})_{1} = 0. Hence (T_{f}^{∞})_{1} = Id from the last diagram. □

(4.8) Corollary: Let <, > be the intersection form on H_{n}(X_{t_{0}}^{0}) (or, by Poincare duality, the form (α, β) ↦ ∫ α ∧ β on H_{c}^{n}(X_{t_{0}}^{0})). Then one has:

dim H_{n}(∂X_{t_{0}}^{0}) = rank <, >= rank(T_{f,c}^{∞}−Id) = rank(T_{f}^{∞}−Id) = μ^{∞}−p_{n−1}(X^{∞}).

In particular, the intersection form depends on the position of the singular points of X^{∞}.

(4.9) Remark: Notice that, similarly as in the local case of isolated hypersurface singularities, the intersection form <, > and the monodromy T_{f}^{∞} can be determined from the variation map V ar. To see this, set H := H_{n}(X_{t_{0}}^{0}, R), denote by b : H → H^{∗} the map given by b(x) =< x, · > and by T_{f,c}^{∞} the monodromy action on H. Then the variation map is a map V ar : H^{∗} → H and (after a canonical identification of H^{∗∗} and H) one has:

T_{f,c}^{∞} = (−1)^{n+1}V ar ∘ (V ar^{∗})^{−1}

T_{f}^{∞} = ((T_{f,c}^{∞})^{∗})^{−1}; and
b = −(V ar)^{−1} − (−1)^{n} ∘ (V ar^{∗})^{−1}.

Actually, all these invariants are defined over the integers, V ar is unimodular and it is equivalent to the Seifert form of the Milnor fibration at infinity (cf. 2.2.v).

§5. The d-th power of the monodromy.

(5.1) In this section we determine the d-th power of T (T being the transformation introduced in §4). Since the Jordan block structure of the unipotent parts of T and T^{d} are the same, this will give much information about the unipotent part of the monodromy at infinity via (4.6).

In the computation of T^{d} there are two (rather different) cases. If λ ≠ 1 then (T^{d})_{λ} is completely local: in (5.3) we describe it in terms of the local transformations T_{i}. The result (and also its proof) is topological. On the other hand, if λ = 1, then (T^{d})_{1} is described Hodge theoretically (cf. 5.5) in terms of local data and the weight filtration (of the mixed Hodge structure) of the d-fold cyclic covering of H^{∞} branched along X_{f}^{∞}.

For this purpose we will introduce a map π′ : X′ → D′, where D′ is a disk in the complex plane, which induces a fibration over the punctured disk with algebraic monodromy T^{−d}, and such that the central fiber has only isolated singularities. The map π′ provides smoothings of these singularities and we will determine the relation between the monodromies of these smoothings and the transformations T_{i}.

(5.2) We start with some preliminary constructions.

As in the previous section, we can assume that f = f_{d} + x_{n+1}^{d−1}. Set:

X_{∞} = {([x], t) ∈ P^{n+1} × D | t · (f_{d}(x_{1}, …, x_{n+1}) + x_{0}x_{n+1}^{d−1}) = x_{0}^{d}}

where D denotes a disk of small radius in the complex plane and let π_{∞} : X_{∞} → D denote the map induced by the projection onto D. The map π_{∞} induces a locally trivial fibration over D − {0} with fiber π_{∞}^{−1}(t) = X_{1/t}, and the algebraic monodromy acting on H^{n}(X_{1/t}) is T^{−1}. Notice that the singular locus of X_{∞} is X^{∞} × {0}.

Consider now the pullback of the map π_{∞} over the map δ : D′ → D defined by δ(t) = t^{d}, D′ being again a disk of small radius. Then X_{∞} ×_{δ} D′ can be identified with

{([x], t) ∈ P^{n+1} × D′ | t^{d} · (f_{d}(x_{1}, …, x_{n+1}) + x_{0}x_{n+1}^{d−1}) = x_{0}^{d}}.

Moreover, the pullback π′_{∞} of π_{∞} is induced by the second projection. Obviously, X_{∞} ×_{δ} D′ over D^{′∗} := D′ − {0} is a fiber bundle with the same fiber as π_{∞}, and with algebraic monodromy T^{−d}.

Consider now the space:

X′ = {([x], t) ∈ P^{n+1} × D′ | f_{d}(x_{1}, …, x_{n+1}) + t · x_{0}x_{n+1}^{d−1} = x_{0}^{d}}

and the map π′ : X′ → D′ induced by the second projection. Then θ : X′ → X_{∞} ×_{δ} D′, given by θ([x], t) ↦ ([tx_{0} : x_{1} : …: x_{n+1}], t) is the normalization map, and identifies the fiber bundles over D^{′∗} induced by π′_{∞} and π′. Denote X′_{s} = (π′)^{−1}(t) for t ∈ D′.

Therefore, we have constructed a map π′ : X′ → D′ which is a fiber bundle over D^{′∗}, with the same fiber as π_{∞} and with algebraic mondromy T^{−d}. Moreover, Sing(X′) = Sing(X′_{0}) is the finite set Sing(X^{∞}) × {0}. The singularities of X′_{0} are the d-th suspensions of the singularities of X^{∞} and the map π′ provides their smoothings.

Set Sing(X′_{0}) = {p′_{1}, …, p′_{k}}, p′_{i} = [0 : p_{i}] and let F′_{i}, T′_{i} denote the Milnor fiber and the algebraic monodromy (acting on H^{n}(F′_{i})) of the smoothing of (X′_{0}, p′_{i}) given by π′.

Now we will formulate the first part of the main result of this section.

(5.3) Theorem: Let H^{n}(F)_{ξ} be the generalized eigenspace with respect to the eigenvalue ξ of T_{f}^{∞}. Then, for any root of unity λ ≠ 1, we can identify ⊕_{ξ^{d}=λ}H^{n}(F)_{ξ} with the generalized λ–eigenspace of ⊕_{i=1}^{k}(H^{n−1}(F_{i})^{⊕(d−1)}) provided by the operator ⊕_{i=1}^{k}c_{d−1}(T^{−d}). By this identification:

[(T_{f}^{∞})^{d}]_{λ} = ⊕_{i=1}^{k}[c_{d−1}(T_{i}^{−d})]_{λ}.

Remark: Recall that for λ ≠ 1 one has: [(T_{f}^{∞})^{d}]_{λ} = [T^{d}]_{λ} (cf. 4.6.a).

Proof of (5.3): From the Leray spectral sequence associated to the specialization map restricted to X′_{t} (for t ≠ 0 and fixed) we get a sequence of vanishing cycles, equivariant with respect to the monodromy:

(10) 0 → H^{n}(X′_{0}) → H^{n}(X′_{t}) → ⊕_{i=1}^{k}H^{n}(F′_{i}) → P^{n+1}(X′_{0}) → 0

where P^{n+1}(X′_{0}) ≃ ker[H^{n+1}(X′_{0}) → H^{n+1}(X′_{t})] denotes the primitive cohomology. In particular, for λ ≠ 1 we have

H^{n}(X′_{t})_{λ} ≃ ⊕_{i=1}^{k}H^{n}(F′_{i})_{λ}.

We want to relate now the action of T′_{i} on H^{n}(F′_{i}) with that of T_{i} on H^{n−1}(F_{i}). This is now a local problem. Fix p_{i} ∈ Sing(X^{∞}) and assume that p_{i} = [0 : …: 0 : 1]. We have to study the (local) smoothing:

Y = {(y_{0}, y, t) | g_{i}(y) + ty_{0} = y_{0}^{d}} → D

given by (y_{0}, y, t) ↦ t. In the sequel D denotes a sufficiently small disc. Consider the map φ : Y → D ×D given by φ(y_{0}, y, t) = (t, y_{0}). Then it is not difficult to verify that φ defines an isolated complete intersection singularity. In the sequel φ : Y → D×D will denote a “good representative” of this icis in the sense of [10]. The discriminant of φ is given by Δ = {y_{0}^{d} = ty_{0}} ⊂ D ×D. Moreover, the following properties hold:

i) Over the complement of Δ, φ is a fiber bundle with fiber F_{i},

ii) φ^{−1}((t, y_{0})) is contractible for (t, y_{0}) ∈ Δ,

iii) the monodromy of the fiber bundle (over D × D − Δ) is abelian, the monodromy induced by a small oriented circle around Δ (constructed in any transversal slice at a smooth point of Δ) is T_{i},

iv) pr_{1} ∘ φ is the local smoothing of X′_{0} given by π′, in particular F′_{i} can be identified with φ^{−1}({t = t_{0}}) for t_{0} ≠ 0 sufficiently small.

Fix t_{0} > 0 sufficiently small. Then {(t, y_{0}) ∈ D × D | t = t_{0}} intersects Δ at the points q_{0} = (t_{0}, 0) and q_{k} = (e^{2πik/d−1} · ^{d−1} \sqrt{t_{0}}, t_{0}), where 1 ≤ k ≤ d − 1. Let I_{k} be the real segment q̄_{0}q_{k} (k=1,…,d-1) and I = ⋃_{k=1}^{d−1} I_{k} ⊂ {t = t_{0}} ⊂ D ×D. Then I is a strong deformation retract of {t = t_{0}} and this retraction can be lifted via φ. In particular, F′_{i} has the homotopy type of φ^{−1}(I). Let r_{k} be the middle point of the segment I_{k}. Since φ^{−1}(q_{k}) is contractible for any k = 0, …, d − 1 and φ^{−1}(r_{k}) ≃ F_{i}, the space φ^{−1}(I_{k}) has the homotopy type of the suspension S(F_{i}) of F_{i}. It follows then that F′_{i} ≃ ⋁_{d−1} S(F_{i}).

In order to compute the monodromy action on ⋁_{d−1} S(F_{i}), notice first that the reduced homology of ⋁_{d−1} S(F_{i}) is generated by the suspension of the cycles of the spaces φ^{−1}(r_{k}). Let t = t_{0}e^{2πiα} , α ∈ [0, 1]. Then the points r_{1}, …, r_{d−1} move along the paths

r_{k}(α) = (t_{0}e^{2πiα}, e^{2πi(α+k)/(d−1)} · ^{d−1} \sqrt{t_{0}}) , α ∈ [0, 1] , 1 ≤ k ≤ d − 1.

We identify the fibers φ^{−1}(r_{k}) (1 ≤ k ≤ d − 1) via the paths r_{k}(α) (1 ≤ k ≤ d − 2). Then, with the notations introduced in (1.2) the monodromy T′_{i} is of the form c_{d−1}(M), where M is induced by the geometric monodromy M_{geom} : φ^{−1}(r_{1}) → φ^{−1}(r_{1}) induced by the loop γ = r_{d−1}(α) ∘ … ∘ r_{1}(α) . Now the linking number of γ with {y_{0} = 0} is one, and with {y_{0}^{d−1} = t} is d − 1. Thus M = T_{i}^{d} and one has:

(T^{−d})_{λ} = ⊕_{i=1}^{k}[c_{d−1}(T_{i}^{d})]_{λ}.

By (4.6), (T^{−d})_{λ} = (T_{f}^{−d})_{λ}. Also, c_{d−1}(T_{i})^{−1} is conjugate to c_{d−1}(T_{i}^{−1}). Thus taking inverses in the above equality the theorem follows. □ (5.4) Example: The theorem above determines, in terms of the local monodromies, the Jordan blocks of T_{f}^{∞} corresponding to eigenvalues ξ, with ξ^{d} ≠ 1. In some cases all the eigenvalues ξ satisfy ξ^{d} ≠ 1, this fact can be verified by the computation of the characteristic polynomial of T_{f}^{∞} (cf. 3.3). In these cases, the above theorem describes completely T_{f}^{∞}.

For example, if f_{d} ∈ C[X, Y, Z] is a product of d linear forms defining d lines in P^{2} intersecting at one point, then the monodromy at infinity of any (∗)-polynomial with highest degree form f_{d} satisfies the above condition. It turns out that T_{f}^{∞} is of finite order, in particular it can be completely determined from (3.3).

As we will see (in the next theorem and in some of the examples of §6), it is not possible to obtain a similar description of the stucture of the Jordan blocks associated to the d-th roots of unity, because this information is not purely local anymore.

(5.5) Theorem: Consider the morphism of mixed Hodge structures:

N := log((T^{−d})_{1}) : H^{n}(X′_{t})_{1} → H^{n}(X′_{t})_{1}(−1).

Recall that X′_{0} is the d-fold cyclic covering of H^{∞} branched along X^{∞}. Then there is an exact sequence of mixed Hodge structures:

0 → H^{n}(X′_{0}) → ker N → ⊕_{i=1}^{k}H_{{p′_{i}}}^{n+1}(X′) → 0.

In particular, for any l one has:

#_{l}(T^{−d})_{1} = dim Gr_{n−l+1}^{W}H^{n}(X′_{0}) + ∑_{i=1}^{k} #_{l}(T_{i})_{1},

and

#(T^{−d})_{1} = dim H^{n}(X′_{0}) + ∑_{i=1}^{k} dim ker(T_{i} − Id).

Proof. The exact sequence is given by Theorem 2 in the Appendix, since sp^{∗} is injective (cf. the exact sequence (10) in the proof of (5.3)). Since the weight filtration on H^{n}(X′_{t})_{1} is the monodromy weight filtration of the nilpotent endomorphism N one has: #_{l}(T^{−d})_{1} = dim Gr_{n−l+1}^{W} ker N.

The local equation defining X′ in a neighborhood of p′_{i} (in C^{n+2}) is ĝ_{i} := g_{i} + x_{0}t − x_{0}^{d} = 0. Let F̂_{i}, T̂_{i} denote the corresponding Milnor fiber and monodromy acting on H^{n+1}(F̂_{i}). It follows from the Sebastiani-Thom formula that there is an isomorphism H^{n+1}(F̂_{i}) ≃ H^{n−1}(F_{i}) compatible with the actions of T̂_{i}, respectively of T_{i}.

From the following exact sequence (cf. [22]):

0 → H_{{p′_{i}}}^{n+1}(X′) → H_{c}^{n+1}(F̂_{i})_{1} →^{j} H^{n+1}(F̂_{i})_{1} → H_{{p′_{i}}}^{n+2}(X′) → 0

one has the identifications: dim Gr_{n−l+1}^{W}H_{{p′_{i}}}^{n+1}(X′) = dim Gr_{n−l+1}^{W} ker j = dim Gr_{n−l+1}^{W} ker Nĝ_{i} = #_{l}(T̂_{i})_{1} = #_{l}(T_{i})_{1}. □

Notice that Theorems (5.3), (5.5) and (4.6) give an almost complete description of the nilpotent (or unipotent) part of T_{f}^{∞}.

The next criterion shows that even in the case λ = 1 the transformation (T^{−d})_{1} can be local if the local transformations T_{i} satisfy some restrictions.

(5.6) Proposition: With the notations previously introduced, assume that one has: a) #_{s}(T_{i})_{1} = 0 for s > 1 and i = 1, …, k.

b) T_{i} has no d-th root of unity different from 1 as eigenvalue for i = 1, …, k.

Then (T^{d})_{1} (acting on H^{n}(X′_{t})_{1}) is the identity. In particular:

a’) #_{k}(T_{f}^{∞})_{1} = 0 for k > 2.

b’) #_{k}(T_{f}^{∞})_{λ} = 0 for λ^{d} = 1, λ ≠ 1 and k > 1. Actually, in this case the monodromy at infinity T_{f}^{∞} can be completely determined from the local monodromies T_{i} and the Betti numbers of X^{∞}. (See (5.3) and the last relation #_{2}(T_{f}^{∞})_{1} = p_{n−1}(X^{∞}) of (4.6).) Proof: From the exact sequence (10) one has that

(11) dim \frac{H^{n}(X′_{t})_{1}}{H^{n}(X′_{0})} = ∑_{i=1}^{k} dim H^{n}(F′_{i})_{1} − dim P^{n+1}(X′_{0}).

The relation T′_{i} = c_{d−1}(T_{i}^{d}) (see the proof of (5.3)), and b) give that dim H^{n}(F′_{i})_{1} = dim H^{n−1}(F_{i})_{1}.

On the other hand, from (5.5) or from Theorem 2 in the Appendix one has:

(12) dim \frac{ker[T^{−d} − Id : H^{n}(X′_{t})_{1} → H^{n}(X′_{t})_{1}]}{H^{n}(X′_{0})} = ∑_{i=1}^{k} dim H_{\{p′_{i}\}}^{n+1}(X′).

Then from a) above and from the Sebastiani–Thom formula one has

dim H_{{p′_{i}}}^{n+1}(X′) = dim(ker[T̂_{i} − Id : H^{n+1}(F̂_{i}) → H^{n+1}(F̂_{i})]) = dim H^{n−1}(F_{i})_{1}.

Now from (11) and (12) we get that (T^{d})_{1} = Id on H^{n}(X′_{t})_{1}, therefore a’) and b’) follow from (4.6).

Notice also that P^{n+1}(X′_{0}) = 0. □

(5.7) Remark: The vanishing of P^{n+1}(X′_{0}) = 0 under assumptions a) and b) above means that the only root of the Alexander polynomial Δ_{X^{∞}}^{1} of the hypersurface X^{∞} is 1 (cf. [9], [5, chapter 6, 3.24]), and then it is Δ_{X^{∞}}^{1}(t) = (t − 1)^{δ} , δ = dim P^{n}(X^{∞}). We recall that Δ_{X^{∞}}^{1} agrees with the characteristic polynomial of the monodromy acting on H^{n−1}(F_{d}), where F_{d} is the Milnor fiber of the map germ f_{d} : (C^{n+1}, 0) → (C, 0), cf. [5].

The above proposition has the following consequence, which is significant on its own, and can be formulated independently of the results of the paper:

(5.8) Corollary: Let Let X^{∞} ⊂ P^{n} be a hypersurface with isolated singularities, and d = deg X^{∞}. Assume that the local monodromies T_{i} of these singular points satisfy:

a) #_{s}(T_{i})_{1} = 0 for s > 1 and i = 1, …, k; and

b) T_{i} has no d-th root of unity different from 1 as eigenvalue for i = 1, …, k.

Then the d-fold covering Y of P^{n} branched along X^{∞} has the following properties:

a’) the primitive cohomology P^{n+1}(Y) = 0, and

b’) the mixed Hodge strucure on H^{n}(Y) is pure of weight n.

§6. Examples.

(6.1) The case n = 1.

Write f_{d} in the form f_{d} = ∏_{i=1}^{m} l^{α_{i}}, where l_{i} are different linear forms. If f is a (∗)-polynomial then by (2.6) its monodromy at infinity is completely determined by the integers α_{i} ∈ N^{∗}. Moreover, f is “good” in the sense of Neumann (cf. [16]) and it has a RPI splice diagram which describes completely the link at infinity of an arbitrary fiber f^{−1}(t) (defined as f^{−1}(t)∩S_{R}, R ≫ 0) and the Waldhausen (splice) decomposition of the link complement S_{R} − f^{−1}(t). This diagram provides the whole set of invariants of the Milnor fibration at infinity (cf. [7], [16]). By the algorithm described in ([16, section 4]) we deduce easily that the splice diagram of f at infinity is:

[figure: •; ⋮; ⋮; ∘; ∘; ∘; ∘; ∘; ∘; (1); (1); (1); α_{1}; α_{i}; α_{m}; α_{1} − 1; α_{i} − 1; α_{m} − 1; 1; 1; 1; 1; 1; 1; (the root vertex is marked “•”).]

It follows (from (3.3) or [7, 11.3]) that the characteristic polynomial of T_{f}^{∞} is:

char_{f}^{∞}(λ) = (λ − 1) · (λ^{d} − 1)^{m−2} · ∏_{i=1}^{m} \frac{λ^{(d−1)α_{i}} − 1}{λ^{d−1} − 1}.

Set q = d(d − 1) ∏_{i=1}^{m} α_{i}. Then λ^{q} = 1 for any root λ of the characteristic polynomial. Then by [7, Theorem 14.1] one has

det((λ · Id − T_{f}^{∞}) |_{Im((T_{f}^{∞})^{q}−Id)}) = \frac{∏_{i=1}^{m}(λ^{d_{i}} − 1)}{(λ − 1)^{m−1}(λ^{α} − 1)}

where d_{i} = gcd(d, α_{i}) and α = gcd(α_{1}, …, α_{m}). It follows that:

• μ^{∞} = d^{2} − 3d + 1 + m,

• #_{2}T_{f}^{∞} = ∑_{i=1}^{m} d_{i} − α − m + 1,

• (T_{f}^{∞})_{1} = Id,

• T_{f}^{∞} is of finite order if and only if d_{i} = 1 for i = 1, …, m.

(6.2) Remark: In the above case, the link of f at infinity can be realized as the link of an isolated curve singularity g : (C^{2}, 0) → (C, 0) if and only if either m = 1 and α_{1} = d or m = d and α_{1} = … = α_{d} = 1. In these cases f can be assumed to be f = x_{1}^{d} + x_{2}^{d−1} or f = x_{1}^{d} + x_{2}^{d} respectively. Exactly in these cases the minimal RPI splice diagrams do not have edges. In all other cases the minimal splice diagram satisfies “reverse Puiseaux inequalities” (cf. [16]), i.e. all “edge determinants” are negative (recall that the “edge determinants” associated with the germ of a plane curve singularity are positive).

(6.3) Lines in general position.

Let l_{1}, …, l_{d} ∈ C[X_{1}, X_{2}, X_{3}] be distinct linear forms defining a set of lines in P^{2} such that no more than two lines meet at a point. Let f be a (∗)-polynomial with highest degree form l_{1} · . . . · l_{d}. It follows from Theorem (5.3) and Proposition (5.6) that T_{f}^{∞} can have Jordan blocks of size bigger than one only for eigenvalue 1. The number of the blocks associated with eigenvalue one can be computed from theorem (4.6), it is (d − 1)(d − 2)/2. The generalized eigenspace of (T_{f}^{∞})_{1} has dimension (d − 1)(d − 2) (from 3.3). Therefore, there are no Jordan blocks of size one associated to the eigenvalue 1. The monodromy at infinity can now be completely determined using Theorem (3.3).

(6.4) Quintic hypersurfaces in P^{4}.

As a first example of the influence of the position of the singularities of X^{∞} on the block structure of the monodromy at infinity, consider the hypersurfaces Y, Z described in ([26, pp. 50 and 55]). Both are quintic hypersurfaces in P^{4} with 118 nodes but b_{4}(Y) = 19 while b_{4}(Z) = 20. Let f_{Y} (resp. f_{Z}) be (∗)- polynomials of degree 5 which have as highest degree form a polynomial defining Y (resp. Z) in P^{4}. Set T_{Y} = T_{f_{Y}}^{∞}, T_{Z} = T_{f_{Z}}^{∞} As in the previous example, T_{Y} and T_{Z} can have Jordan blocks of size bigger than one only for eigenvalue 1. The number of the blocks of size two can be computed using Theorem (4.6) and it turns out to be 100 for Y and 99 for Z (dim H^{4}(X_{t_{0}}^{0})_{1} = 204 in both cases). Combining this with Theorem (3.3) we can completely describe T_{Y} and T_{Z}.

(6.5) Zariski’s sextics.

Let f_{6} ∈ C[X_{1}, X_{2}, X_{3}] be a form defining a plane sextic in P^{2} with six cusps and no other singularities. Let f be any (∗)-polynomial with highest degree form f_{6}. Then from (4.7) and (5.3) it follows that T_{f}^{∞} has no Jordan blocks of size bigger than one associated neither to the eigenvalue 1 nor to eigenvalues λ with λ^{6} ≠ 1. Also, from the sequence (10), the computation of the action of monodromy on ⊕_{i=1}^{k}H^{n}(F′_{i}) in the proof of (5.3) and from (4.6.a), it follows that T_{f}^{∞} can have Jordan blocks of size at most two. The number of them depends on the position of the cusps, more precisely:

• #_{2}T_{f}^{∞} = 10 if the six cusps are on a conic,

• #_{2}T_{f}^{∞} = 12 if they are not.

Because from the computation of the characteristic polynomial of T_{f}^{∞} in §2 and the sequence in (4.4) one gets that

dim H^{2}(X′_{t})_{1} = ∑_{λ^{6}=1} dim H^{2}(X_{t})_{λ} = 58,

(where X′_{t} is defined in the proof of (5.3)). On the other hand, from Theorem (5.5) one gets that:

#(T^{−6})_{1} = dim H^{2}(X′_{0}) = 46 + δ

where δ = dim H^{3}(X′_{0}), X′_{0} being the hypersurface in P^{3} defined by x_{0}^{6} = f_{6}, i.e., the 6-fold cyclic covering of P^{2} branched along the curve X^{∞}. The possible values of δ are known to be 2 if the six cusps are on a conic or 0 if they are not (cf. [27, VIII, §3]), and then the result follows.

Acknowledgements. This paper was written during a stay both authors made at the University of Nijmegen. We are grateful to the staff of the Mathematical Institute for their hospitality. We would like to thank especially to J. Steenbrink his continuous encouragement and useful conversations.

APPENDIX: On the local invariant cycle theorem by R. García López and J.H.M. Steenbrink

In this note all cohomology groups will be assumed to have coefficients in the field Q of rational numbers. We prove the following two theorems: Theorem 1: Let X be a complex analytic space which can be embedded in a projective variety as an open analytic subset. Let π : X → D be a flat projective holomorphic map onto the unit disk D in the complex plane. Let Z be the singular locus of X, set Y = π^{−1}(0) and assume that Z ⊂ Y. Let X_{t} be the generic fiber of π. Let k ∈ N and let T ∈ Aut(H^{k}(X_{t})) be the monodromy transformation of π around the critical value 0. Then the sequence

H^{k}(X − Z) → H^{k}(X_{t}) ⟶^{T −Id} H^{k}(X_{t})

is exact. Remarks: 1. The first map in the sequence above is the restriction map.

2. If Z = ∅, the theorem is due to Katz in the setting of l-adic cohomology and to Clemens and Schmid in the Kähler case ([3]).

3. The hypothesis Z ⊂ π^{−1}(0) is equivalent to the generic fiber of π being smooth. Proof. After possibly shrinking D, we may assume that the restriction of π over the punctured disk D − {0} is a C^{∞}- fiber bundle and that the inclusion Y ↪ X is a homotopy equivalence. Let then X̃ be the limit fiber of π, defined as X̃ = X ×_{D} H, where H is the universal covering space of D − {0}. We recall that X_{t} and X̃ are of the same homotopy type. In the sequence

H^{k}(X − Z) ⟶^{α} H^{k}(X − Y) ⟶^{β} H^{k}(X̃)

one has Im(β) = Ker(T − Id) by the Wang sequence. The terms in this sequence carry mixed Hodge structures (MHS) such that α and β become morphisms of MHS. We use Saito’s formalism of mixed Hodge modules ([18]).

• For H^{k}(X̃) one has the limit MHS ([20], [23]) given by H^{k}(X̃) ≃ H^{k}(Y, Ψ_{f}Q_{X}^{H}).

• Let C ⊂ Y be any closed analytic subset, let i : Y ↪ X and j : X − C ↪ X be the inclusion maps. Then

H^{k}(X − C) ≃ H^{k}(Y, i^{∗}Rj_{∗}j^{∗}Q_{X}^{H})

gives H^{k}(X − C) a MHS.

By [20], Ker(T −Id) has weight ≤ k. Hence it suffices to show that W_{k}H^{k}(X− Y) = α(W_{k}H^{k}(X − Z)), where W_{•} denotes the corresponding weight filtration. One has the exact sequence of MHS

H^{k}(X − Z) → H^{k}(X − Y) → H^{k+1}(X − Z, X − Y).

Fix a projective variety W containing X as an open analytic subset. Without loss of generality we can assume that W − Z is smooth. By excision we have an isomorphism of MHS H^{k+1}(W − Z, W − Y) ≃ H^{k+1}(X − Z, X − Y). We also have the exact sequence of MHS

H^{k}(W − Z) → H^{k}(W − Y) → H^{k+1}(W − Z, W − Y) → H^{k+1}(W − Z).

Now W_{k}H^{k+1}(W − Z) = 0 as W − Z is smooth, moreover W_{k}H^{k}(W − Z) = Im(H^{k}(W) → H^{k}(W − Z)) and similarly for W_{k}H^{k}(W − Y), so W_{k}H^{k}(W − Z) → W_{k}H^{k}(W − Y) is surjective. We conclude that W_{k}H^{k+1}(W − Z, W − Y) = 0. Hence α : W_{k}H^{k}(X − Z) → W_{k}H^{k}(X − Y) is surjective. □

Remark: M. Saito has informed us that the theorem above follows also from the results in [19]. Actually, if IH^{∗}(X) denotes the intersection cohomology of X then, with the notations above one has a factorization

IH^{k}(X) → H^{k}(X − Z) → H^{k}(X_{t})

and Theorem 1 follows then from [19, (3.8)]. If the central fiber has only isolated complete intersection singularities (icis) then we have: Theorem 2: In addition to the hypothesis of Theorem 1 and with the same notations, assume that Y = π^{−1}(0) has only icis and set dim(X) = n + 1. Then there is an isomorphism:

\frac{ker [T − Id : H^{n}(X_{t}) → H^{n}(X_{t})]}{im [sp^{∗} : H^{n}(Y) → H^{n}(X_{t})]} ≃ H_{Z}^{n+1}(X).

where sp^{∗} denotes the morphism induced in cohomology by the specialization map. Remarks: 1) The isomorphism above is also an isomorphism of mixed Hodge structures.

2) In the applications in §§5-6 of the paper above, X is a hypersurface with isolated singularities. Given p ∈ Z, let g_{p} : (C^{n+2}, 0) → (C, 0) be a map germ defining the germ (X, p) and let F_{p}, T_{p} be the corresponding Milnor fiber and local monodromy acting on H^{n+1}(F_{p}). Then we recall that there is an isomorphism:

H_{{p}}^{n+1}(X) ≃ coker[T_{p} − Id : H^{n+1}(F_{p}) → H^{n+1}(F_{p})].

Proof. We claim first that there is an isomorphism W_{n}H^{n}(X−Z) ≃ W_{n}H^{n}(X− Y). One can prove as in the proof of Theorem 1 that W_{n}H^{n+1}(X − Z, X − Y) = 0, so from the exact sequence of the pair (X − Z, X − Y) it follows that in order to prove the claim it is enough to show that the map H^{n−1}(X − Y) → H^{n}(X − Z, X − Y) is surjective. Since the singularities of Y are icis, it follows from the long exact sequence of vanishing cycles that the monodromy acts as the identity on H^{k}(X̃) for k ≠ n. Assume that n ≥ 2. Then the map above fits in a commutative diagram with exact row:

[figure: H^{n−1}(X − Z); H^{n−1}(X − Y); H^{n}(X − Z, X − Y); H^{n−2}(X̃)(−1); γ]

and the MHS of H^{n−2}(X̃)(−1) is pure of weight n. Since the singularities of the total space X are also icis, we have that H^{n−1}(X − Z) ≃ H^{n−1}(X) ≃ H^{n−1}(Y) and since Y is complete the weights of H^{n−1}(Y) are ≤ n − 1. It follows then that the map γ above is injective. On the other hand, one has isomorphisms:

H^{n}(X − Z, X − Y) ≃ H^{n−2}(Y − Z)(−1) ≃ H^{n−2}(Y)(−1) ≃ H^{n−2}(X̃)(−1)

The first is a Thom isomorphism, the second follows from the fact that the singularities of Y are icis (so H_{Z}^{n−2}(Y) = H_{Z}^{n−1}(Y) = 0) and the third is induced by the specialization map. So dim H^{n}(X − Z, X − Y) = dim H^{n−2}(X̃), thus γ is an isomorphism and the claim follows. The case n = 1 is similar and left to the reader. Since Y ↪ X is a homotopy equivalence, from the exact sequence of the couple (X, X − Z) we get the exact sequence:

H^{n}(Y) ⟶^{δ} W_{n}H^{n}(X − Z) → W_{n}H_{Z}^{n+1}(X) → W_{n}H^{n+1}(Y).

Since the singularities of Y and X are isolated, it follows from [24], [12] that W_{n}H^{n+1}(Y) = 0 and W_{n}H_{Z}^{n+1}(X) ≃ H_{Z}^{n+1}(X). So we have:

[figure: 0; W_{n}H^{n}(X − Z); H^{n}(X̃); H^{n}(X̃); H^{n}(Y); sp^{∗}; T − Id; T − Id; δ]

with coker(δ) ≃ H_{Z}^{n+1}(X). The horizontal sequence comes from the Wang sequence and is exact by the claim above and the fact that the weights of ker(T − Id) are ≤ n. The theorem follows then from an easy diagram-chase. □

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Barcelona, Spain. e-mail address: rgarcia@cerber.ub.es

The Ohio State University, 231 West 18th Avenue, Columbus OH 43210, USA. e-mail address: nemethi@math.ohio-state.edu