Hodge numbers attached to a polynomial map

by

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

1. Introduction

(1.1) In the last years the behaviour at infinity of polynomial maps has been extensively studied, basically by means of topological invariants: Eisenbud-Neumann diagrams, monodromy at infinity, etc. In order to carry further this analysis of the asymptotic behaviour of polynomial maps we consider in this paper the following approach: One can attach to any polynomial map a geometrical variation of mixed Hodge structures (from now on, MHS; see (1.2) below for details). Using either the geometric methods of [24], [9] or Saito’s theory of mixed Hodge modules ([17]), one gets that this variation gives rise to a functorial limit MHS. The equivariant Hodge numbers of this limit MHS are analytical invariants of the polynomial map under consideration and in this paper we compute them for a class of polynomials which was extensively studied in [5], [6] from a topological point of view. More precisely, we determine the equivariant Hodge numbers of this limit MHS in terms of equivariant Hodge numbers attached to some isolated hypersurface singularities and Hodge numbers of cyclic coverings of projective space branched along a hypersurface (see theorem (3.1) for the precise statement). Both these local and global Hodge numbers can be explicitly computed in a number of cases, as shown in the examples in section 7.

(1.2) We give the precise description of the invariants we are going to study. Let f : ℂ^{n+1} → ℂ be a polynomial map. It is known that there is a finite set Γ_{f} such that f defines a locally trivial C^{∞}-fibration over ℂ − Γ_{f} (cf. e.g. [15, Appendix]). Given a positive number r ∈ ℝ, denote by D_{r} the disk in the complex plane of center 0 and radius r, and set D_{r}^{∗} = D_{r} − {0}, Z_{r} = ℂ^{n+1} − f^{−1}(D_{r}). If r > max{ |x| : x ∈ Γ_{f} } then the map

1Partially supported by the DGCYT, PB95-0274 and INTAS 97-1644.

2Partially supported by NSF Grant No. DMS-9622724.

1/f : Z_{r} ⟶ D_{1/r}^{∗}

is a locally trivial C^{∞}-fibration. Actually we have a projective system of equivalent fibrations indexed by r, thus we can assume r so big as necessary. Now we compactify the map 1/f and we add a fiber over 0 ∈ ℂ. That is, we take an analytical manifold X where Z = Z_{r} is embedded as an open dense subset, (X − Z)_{red} is a divisor with normal crossings and smooth irreducible components and the map 1/f extends to a flat projective map p : X ⟶ D_{1/r}, smooth over D_{1/r}^{∗}. Set X −Z = Y ∪Δ, where Y = p^{−1}(0) and Δ is the union of the irreducible components of X − Z not in Y. Increasing r if necessary, we can assume that the restriction of p to any intersection of components of Δ_{red} is smooth, so that Δ becomes a relative divisor with normal crossings. Let ℍ be the universal covering space of D_{1/r}^{∗}, set Z̃ = Z ×_{D_{1/r}} ℍ. Notice that Z̃ has the homotopy type of a generic fiber of f. Then the cohomology groups H^{i}(Z̃, ℚ) carry a limit MHS. This follows for example from [24, §5] or from [9, Théorème 5.13]. All MHS we will work with will be regarded as ℚ-MHS. One can prove using standard arguments that this MHS does not depend on the chosen compactification and is functorial on f (and therefore it is for example an invariant of polynomials with respect to algebraic changes of variables in ℂ^{n+1}).

(1.3) Definition: Given a polynomial map f : ℂ^{n+1} ⟶ ℂ, the limit MHS on H^{n}(Z̃, ℚ) obtained in the way described above will be called the MHS at infinity of f.

(1.4) The MHS at infinity should be viewed as a global analogue of the limit MHS associated to a hypersurface singularity in [22] and [9]. If f is a cohomologically tame polynomial (see [16]) there is another definition of a MHS at infinity due to C. Sabbah. For these polynomials, Sabbah has proved that the equivariant Hodge numbers of the two MHS’s are the same. Unless otherwise stated, all (co)homology groups appearing in this paper will be assumed to be reduced and to have complex coefficients. We choose once and for all a root i of −1 in ℂ (equivalently, an orientation on the complex plane) and from now on we will use the notation e(r) := e^{2πir}, r ∈ ℝ. The authors thank Professor C. Sabbah for his help, encouragement and fruitful discussions about Hodge theory. The second author thanks also the warm hospitality of the Department of Mathematics of the Ecole Polytechnique at Palaiseau during a stay he made there.

2. The main construction and exact sequences

(2.1) Although the MHS at infinity is defined for any polynomial map, if one does not place some kind of restriction on f even the topological behaviour at infinity can be very complicated. We will consider the class of polynomials f : ℂ^{n+1} ⟶ ℂ which satisfy the following condition:

(∗) \{ \begin{matrix}If t ∈ ℂ is not a critical value of f, \\ then the closure in ℙ^{n+1} of the affine \\ hypersurface \{f = t\} is non-singular. \end{matrix}

Polynomials satisfying this condition will be called (∗)-polynomials. In this section we recall some consequences of the (∗) condition and some exact sequences attached to a (∗)-polynomial. For details, see [5], [6]. Let f : ℂ^{n+1} ⟶ ℂ be a (∗)-polynomial. Then:

(a) Any fiber of f has at most isolated singularities and has the homotopy type of a bouquet of n-dimensional spheres. Consider the fibration at infinity of f:

f : f^{−1}(∂ D̄_{r}) ∩ B_{R} ⟶ ∂ D̄_{r} ,

where D_{r} (resp. B_{R}) is a disc in ℂ (resp. a ball in ℂ^{n+1}) of radius r (resp. R) and 0 << r << R. This fibration is equivalent to the fibration f : f^{−1}(∂ D̄_{r}) ⟶ ∂ D̄_{r}. For t ∈ ∂ D̄_{r} set X_{t}^{0} = f^{−1}(t) ⊆ ℂ^{n+1}. The geometric monodromy of this fibration (where ∂ D̄_{r} is positively oriented) is denoted by T_{geom}. In [5, 6] we proved some basic properties of the classical algebraic monodromy T_{f}^{∞} : H^{n}(X_{t}^{0}) ⟶ H^{n}(X_{t}^{0}) which is defined by T_{f}^{∞}[ω] = [T_{geom}^{∗}(ω)]. Let X_{t} denote the projective closure of the affine hypersurface X_{t}^{0}. The classical algebraic monodromy over ∂ D̄_{r} on the n-th cohomology groups of the projective closures will be denoted by T : H^{n}(X_{t}) ⟶ H^{n}(X_{t}) (and is defined also as a pull–back).

(b) For R >> 0, the Milnor fibration at infinity φ = \frac{f}{∥f∥} : ∂ B̄_{R} − f^{−1}(0) ⟶ S^{1} exists and it is equivalent to the fibration at infinity of f. In particular, there is a well defined variation map V ar : H^{n}(X_{t}^{0}) ⟶ H_{c}^{n}(X_{t}^{0}) (V ar([ω]) := [T_{geom}^{∗}(ω) − ω]), which is an isomorphism and has the usual compatibility properties with the monodromy T_{f}^{∞}.

(c) The projective hypersurface X^{∞} := {f_{d} = 0} ⊆ ℙ^{n} has only isolated singularities. The semisimple part of T_{f}^{∞} depends only on the local topological type of these singular points, but the nilpotent part of T_{f}^{∞} (and also the intersection form on H_{n}(X_{t}^{0})) depends also on the position of the singularities of X^{∞}. In general, if one wants to relate the algebraic monodromy with Hodge theory, it is more convenient to use the algebraic monodromy induced on the (horizontal sections of the) cohomological Gauss–Manin connection (see e.g.

[1] §12). This is the inverse dual of the classical homological monodromy [loc. cit 13.1.A]. In this paper they will be denoted by M_{f}^{∞} = (T_{f}^{∞})^{−1} acting on H^{n}(X_{t}^{0}), M = T^{−1} acting on H^{n}(X_{t}); and we will call them simply (algebraic) monodromies. Obviously, W ([ω]) := [(T_{geom}^{∗})^{−1}(ω) − ω] has the well–known compatibility with M_{f}^{∞} (see 2.3). We will use the corresponding similar notations in all local situations which will appear in the body of the paper. (2.2) In order to study the MHS at infinity of a (∗)-polynomial we will consider a concrete compactification of the map 1/f (see [5], [6]).

Let f = f_{d} + f_{d−1} + … denote the decomposition of f into homogeneous components, let D denote a disc in the complex plane with center at the origin and sufficiently small radius. Set

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

where [z_{0} : z_{1} : …: z_{n+1}] are homogeneous coordinates in ℙ^{n+1}. The map π : X ⟶ D given by π([z], t) = t induces a locally trivial C^{∞} filtration over D − {0} and the fibers of π are exactly the projective closures of the fibers of f. The monodromy of π over ∂D (with positive orientation) is M. Then by [5] we have a commutative diagram

0 → P_{X^{∞}}^{n}(X_{t}) → P^{n}(X_{t}) → H^{n}(X_{t}^{0}) → P_{X^{∞}}^{n+1}(X_{t}) → 0
(E.0) ↓ id ↓ M ↓ M_{f}^{∞} ↓ id
0 → P_{X^{∞}}^{n}(X_{t}) → P^{n}(X_{t}) → H^{n}(X_{t}^{0}) → P_{X^{∞}}^{n+1}(X_{t}) → 0

where P denotes primitive cohomology (i.e. for X ⊂ ℙ^{N} one has P^{k}(X) = Coker(H^{k}(ℙ^{N}) → H^{k}(X))). The above spaces carry natural MHS’s: P^{n}(X_{t}) carries Schmid’s MHS, H^{n}(X_{t}^{0}) the MHS introduced in section 1 and all arrows are morphisms of MHS’s.

The compactification provided by π is not such a good one since its center fiber π^{−1}(0) is not reduced (it has multiplicity d) and X is very singular. Therefore, we will consider the normalization of the d-fold covering of π. More precisely, let D′ be again a small disk, δ : D′ ⟶ D the map given by δ(t) = t^{d}. Define X′ as the normalization of X ×_{δ} D′ and let π′ : X′ ⟶ D′ be the natural projection. The map π′ gives a fibration over D′ − {0} with the same fiber as π and classical monodromy T^{−d}. Both X′ and the central fiber X′_{0} = (π′)^{−1}(0) have only isolated singularities. Set Sing X^{∞} = {p_{1}, …, p_{k}} and let F_{j}, M_{j} be the local Milnor fiber and the algebraic monodromy M_{j} : H^{n−1}(F_{j}) ⟶ H^{n−1}(F_{j}) of the hypersurface singularity (X^{∞}, p_{j}) ⊂ (ℙ^{n}, p_{j}) for 1 ≤ j ≤ k (Here ℙ^{n} is the hyperplane at infinity).

Then Sing X′ = Sing X^{∞} × {0} and the central fiber X′_{0} is the d-fold cyclic covering of ℙ^{n} branched along X^{∞}. In particular, if we set Sing X′_{0} = {p′_{1}, …, p′_{k}} then the isolated singularities (X′_{0}, p′_{j}) are the d-th suspensions of the singularities (X^{∞}, p_{j}). Let F′_{j} (resp. M′_{j}) be the Milnor fiber (resp. the monodromy) of the local smoothing π′ : (X′, p′_{j}) ⟶ (D′, 0). We remark that this smoothing is not the (natural) smoothing of (X′_{0}, p′_{j}) with total space smooth. The following commutative diagram is the exact sequence of vanishing cycles corresponding to π′ together with the monodromy action. It is a diagram of MHS’s (cf. [5, (10)] or [6, (2.2)]).

0 → P^{n}(X′_{0}) → P^{n}(X_{t}) → ⨁_{j=1}^{k} H^{n} (F′_{j}) → P^{n+1}(X′_{0}) → 0
(E.1)  ↓ id  ↓ M^{d}  ↓ ⊕ (M′_{j})^{−1}  ↓ id
0 → P^{n}(X′_{0}) → P^{n}(X_{t}) → ⨁_{j=1}^{k} H^{n}(F′_{j}) → P^{n+1}(X′_{0}) → 0.

Notice that X′_{0} ⊂ ℙ^{n+1} is given by the equation z_{0}^{d} = f_{d}(z_{1}, …, z_{n+1}). There is a Galois action of ℤ/dℤ on X′_{0} generated by the automorphism [z_{0} : …: z_{n+1}] ⟼ [e(1/d) z_{0} : …: z_{n+1}] which induces at the cohomology level an automorphism which will be denoted G^{q} : P^{q}(X′_{0}) ⟶ P^{q}(X′_{0}).

The disadvantange of (E.1) is that the monodromy action at the level of P^{n}(X_{t}) is M^{d} instead of M. In [6] we constructed an automorphism of X′ over D′ which is a “lifting” of the geometric monodromy of π. In [loc. cit.] we identified (H^{n} (F′_{j}), M′_{j}) with

(H^{n−1} (F_{j})^{⊕ (d−1)}, c_{d−1} (M_{j})^{d})

where for an operator φ : V ⟶ V and an integer ℓ ≥ 1 we denote c_{ℓ}(φ) : V^{⊕ℓ} ⟶ V^{⊕ℓ} the operator defined by c_{ℓ}(φ) (x_{1}, …, x_{ℓ}) = (φ(x_{ℓ}), x_{1}, …, x_{ℓ−1}). Now the “lifted monodromy action” induces the following commutative diagram (of vector spaces):

0 → P^{n}(X′_{0}) → P^{n}(X_{t}) → ⨁_{j=1}^{k} H^{n−1} (F_{j})^{⊕ (d−1)} → P^{n+1}(X′_{0}) → 0
(E.1’)  ↓ G^{n}  ↓ M  ↓ ⊕ c_{d−1} (M_{j}^{−1})  ↓ G^{n+1}
0 → P^{n}(X′_{0}) → P^{n}(X_{t}) → ⨁_{j=1}^{k} H^{n−1}(F_{j})^{⊕(d−1)} → P^{n+1}(X′_{0}) → 0.

(In [loc. cit.] it is not clarified the MHS-meaning of this diagram, this will be done in the next section. That discussion will show that (E.1’) is actually a diagram of MHS’s. One should be aware of the fact that if we consider on H^{n}(F′_{j}) and H^{n−1}(F_{j}) their natural MHS, then H^{n}(F′_{j}) ≠ H^{n−1}(F_{j})^{⊕(d−1)} as MHS’s.) Theorem 2 in the Appendix of [5] together with the “lifted monodromy action” gives the following commutative diagram of MHS (cf. [6]):

0 → P^{n}(X′_{0}) → Ker (M^{d} − Id | P^{n}(X_{t}) ) → ⨁_{j=1}^{k} H_{{p′_{j}}}^{n+1} (X′) → 0
(E.2)  ↓ G^{n}  ↓ M  ↓ id
0 → P^{n}(X′_{0}) → Ker (M^{d} − id | P^{n}(X_{t}) ) → ⨁_{j=1}^{k} H_{{p′_{j}}}^{n+1} (X′) → 0.

(2.3) There is a nice connection between the variation map and the diagram (E.0) (for details, see [5, pp. 217–218]). There is an orthogonal decomposition P^{n}(X_{t})_{1} = P_{X^{∞}}^{n} (X_{t}) ⊕ P_{X^{∞}}^{n} (X_{t})^{⊥}, M_{1} = Id ⊕ M′ (where P^{n}(X_{t})_{1} denotes the space of generalized eigenvectors of P^{n}(X_{t}) of eigenvalue 1 with respect to M, we will use the same notation for the other groups as well). We identify P_{X^{∞}}^{n}(X_{t})^{⊥} with the image of P^{n} (X_{t}) ⟶ H^{n} (X_{t}^{0}). Then the composed map

w : H^{n} (X_{t}^{0})_{1} ⟶^{W_{1}} H_{c}^{n} (X_{t}^{0})_{1} ⟶ H^{n} (X_{t}^{0})_{1}

(where W was defined in (2.1)) has range exactly P_{X^{∞}}^{n}(X_{t})^{⊥} and makes the following diagram (of vector spaces) commutative

0 → P_{X^{∞}}^{n}(X_{t})^{⊥} →  H^{n}(X_{t}^{0})_{1}  → P_{X^{∞}}^{n+1}(X_{t}) → 0

w↙ ↓ M′ − id ↙ ↓ (M_{f}^{∞})_{1} − id  ↓ 0

0 → P_{X^{∞}}^{n}(X_{t})^{⊥} →  H^{n}(X_{t}^{0})_{1}  → P_{X^{∞}}^{n+1}(X_{t}) → 0.

3. The mixed Hodge structure at infinity in the (∗)-case.

In this section we describe the equivariant Hodge numbers of the MHS at infinity of a (∗)-polynomial. Let f be a (∗)-polynomial and keep the notations of section 2. Let (M_{f}^{∞})_{ss} (resp. (M_{f}^{∞})_{u}) denote the semisimple (resp. the unipotent) part of the monodromy at infinity M_{f}^{∞}. We recall ([9, Théorème 15]) that (M_{f}^{∞})_{ss} is a MHS-automorphism of H^{n}(X_{t}^{0}) of type (0, 0) and N^{0} = log (M_{f}^{∞})_{u} is a MHSautomorphism of type (−1, −1). We will denote H^{n}(X_{t}^{0})_{ξ} the ξ-eigenspace of H^{n}(X_{t}^{0}) with respect to M_{f}^{∞}. For ξ a d-th root of unity, we denote by P^{ℓ}(X′_{0})_{ξ} the ξ-eigenspace of P^{ℓ}(X′_{0}) with respect to the Galois action G^{ℓ}. Obviously, this eigenspace decomposition is compatible with the MHS of P^{ℓ}(X′_{0}). We recall the definition of primitive spaces with respect to a nilpotent endomorphism. (cf. e.g. [10, (5.3)], [19, (6.4)]): Let H be a finitely dimensional ℚ-vector space and N : H → H a nilpotent endomorphism of H. Let W denote the weight filtration on H corresponding to N and centered at m. Assume there is a fitration F on the complexification of H such that (H, W, F) is a MHS. For r ≥ 0 one sets:

P_{r} := KerN^{r+1} : Gr_{m+r}^{W}H → Gr_{m−r−2}^{W}H.

Then P^{r} carries a MHS of weight m + r. For p + q ≥ m we will call P_{p+q−m}^{p,q} the primitive space of type (p, q).

The local vanishing cohomology H^{n−1}(F_{j}) of (X^{∞}, p_{j}) has a MHS (cf. [22]). For ξ ≠ 1 (resp. for ξ = 1) the weight filtration of the generalized ξeigenspace is the monodromy weight filtration centered at n − 1 (resp. at n). In this case, we will denote the dimension of the corresponding primitive spaces (with respect to the infinitesimal nilpotent monodromy) by p_{ξ}^{pq}(X^{∞}, p_{j}). The main theorem of this section is the following:

(3.1) Theorem:

a) The weight filtration of H^{n}(X_{t}^{0})_{1} is the monodromy weight filtration given by N_{1}^{0} with center n+1. The dimensions of the primitive spaces of type (p, q) (p + q ≥ n + 1) are:

p_{1}^{p,q}(f) = h^{n−q, n−p} (P^{n−1}(X^{∞}) ).

b) The weight filtration of H^{n}(X_{t}^{0})_{≠1} is the monodromy weight filtration given by N_{≠1}^{0} with center n. The dimensions of the primitive spaces of type (p, q) are given as follows:

i) If ξ^{d} = 1 and ξ ≠ 1, then

p_{ξ}^{p,q}(f) = h^{n−q, n−p} (P^{n}(X′_{0})_{ξ}).

ii) If ξ^{d} ≠ 1, then

p_{ξ}^{p,q}(f) = ∑_{j=1}^{k} p_{ξ^{1−d}}^{p−[β+γ], q−δ+[β+γ]} (X^{∞}, p_{j})

where ξ = e^{−2πiβ}(0 < β < 1), γ = {(d − 1) β}, and δ = 0 if γ = 0 and δ = 1 if γ ≠ 0. ( Here [ ] denotes integer part and for α ∈ ℝ, {α} = α − [α] ).

Notice that for ξ^{d} ≠ 1, p_{ξ}^{p,q}(f) depends only on the type of the local singularities (X^{∞}, p_{j})_{j=1}^{k} (and not on their position).

(3.2)Remarks: a) The equivariant Hodge numbers h_{λ}^{a,b} of H^{n}(X_{t}^{0}) can be computed from the formula

h_{λ}^{a,b} = ∑_{ℓ≥0} p_{λ}^{a+ℓ, b+ℓ} (f)

where a+b ≥ center of the weight filtration := c, and from h_{λ}^{a,b} = hc−a, c−b ; λ̄ = h_{λ}^{c−b, c−a} for a + b ≤ c (cf. [10, (5.3)]).

b) The above theorem shows that the weight filtration of H^{n}(X_{t}^{0}) has a behaviour similar to that of the weight filtration of the MHS on the vanishing cohomology corresponding to an isolated hypersurface singularity (namely, it is centered at n for λ ≠ 1 and at n + 1 for λ = 1). For λ ≠ 1 this can be proved using Schmid’s results [19] and for λ = 1 follows basically from the fact that the variation map is an isomorphism.

c) It follows also from the theorem that the maximal size of a Jordan block of (M_{f}^{∞})_{ξ} is n + 1 if ξ^{d} = 1 and ξ ≠ 1, otherwise is n (cf. [6, p. 4]).

d) If f and f′ are two (∗)-polynomials with the same highest degree form, their equivariant Hodge numbers are the same.

Proof of theorem 3.1: We will say that two MHS’s H_{1} and H_{2} are numerically equivalent (and we will denote this H_{1} =^{n} H_{2}) if their Hodge numbers are the same. The primitive decomposition theorem gives the following fact, which will be used several times.

(3.3) Lemma: Let H_{i} (i = 1, 2) be a MHS with a nilpotent (−1, −1)– morphism N_{i} such that the weight filtration of H_{i} is the monodromy weight filtration of N_{i} (with the same fixed center). If KerN_{1} =^{n} KerN_{2}, then H_{1} =^{n} H_{2}.

We divide the proof of (3.1) in the following cases. Case ξ = 1. In order to prove this case we study first the MHS’s of P^{∗}(X^{∞}) and P_{X^{∞}}^{∗}(X_{t}). (Some of the results of this subsection will be used in section 4 too.)

(3.4) Lemma: P_{X^{∞}}^{n+i}(X_{t}) and P^{n−i}(X^{∞}) are dual MHS’s with respect to ℚ(−n). P^{n}(X^{∞}) is pure of weight n and P^{n−1}(X^{∞}) = W_{n−1} P^{n−1}(X^{∞}).

Proof of the lemma: The duality statement follows quite directly from Saito’s formalism: On the category of mixed Hodge modules one can define a duality functor which commutes with the nearby cycle functor up to a Tate twist (cf. [17], 2.6) and extends to a functor defined in the derived category, compatible with Verdier duality. In Saito’s terminology and with the notations of (1.2), if i : Δ ↪ X is the inclusion map one has to apply this duality to the object ℝΨ_{p} i_{∗} i^{!} ℚ_{X}[n + 1] in the derived category of mixed Hodge modules. The purity of P^{n}(X^{∞}) is proved in [22, (4.7)]. (3.5) Consider now a 1-dimensional deformation of X^{∞} ⊆ ℙ^{n} by hypersurfaces, i.e. a family of hypersurfaces X_{s}^{∞} ⊆ ℙ^{n} of degree d such that X_{0}^{∞} ≃ X^{∞}, X_{s}^{∞} is smooth for s ≠ 0, and the total space of the deformation is smooth as well. Consider the exact sequence of vanishing cycles

(3.6) 0 → P^{n−1}(X^{∞}) → P^{n−1}(X_{s}^{∞})_{1} → ⊕_{j}H^{n−1}(F_{j})_{1} → P^{n}(X^{∞}) → 0 .

Here P^{∗}(X^{∞}) carries Deligne’s MHS, P^{n−1}(X_{s}^{∞})_{1} Schmid’s MHS and H^{n−1}(F_{j})_{1} Steenbrink’s MHS. Moreover, the Hodge numbers of H^{n−1}(F_{j})_{1} does not depend on the choice of the smoothing (because in a μ–constant family of hypersurface singularities, the equivariant Hodge numbers are constant), in particular, they are equal to the Hodge numbers provided by the natural local smoothing. The sequence is compatible with the monodromy action, which is the identity on P^{∗}(X^{∞}), M_{j} on H^{n−1}(F_{j}) and it will be denoted M_{gℓ} on P^{n−1}(X_{s}^{∞})_{1}. By [19] (resp. [22]) the weight filtration of P^{n−1}(X_{s}^{∞})_{1} (resp. H^{n−1}(F_{j})_{1}) is the monodromy weight filtration with center n−1 (resp. n). By the invariant cycle theorem P^{n−1}(X^{∞}) = Ker (M_{gℓ} − Id). From these facts it follows that we have, for ℓ ≥ 1

dim Gr_{n−1−ℓ}^{W} P^{n−1}(X^{∞}) + dim Gr_{n−1−ℓ}^{W} ⊕_{j} H^{n−1}(F_{j})_{1}

(3.7)  = dim Gr_{n−1+ℓ}^{W} ⊕_{j} H^{n−1}(F_{j})_{1} − dim Gr_{n−1+ℓ}^{W} P^{n}(X^{∞}).

Now we compare ⊕_{j}H^{n−1}(F_{j})_{1} and P^{n}(X_{t})_{1}. The 1-eigenspaces in (E.2) provide the following MHS-identification (notice that G^{n} has no 1–eigenvector on P^{n}(X′_{0})):

(3.8)  Ker (M − Id | P^{n}(X_{t})_{1}) = ⊕_{j=1}^{k} H_{{p′_{j}}}^{n+1} (X′).

Let g_{i} be a local equation defining the singularity germ (X^{∞}, p_{j}). The smoothings of the singularities (X′, p′_{j}) are given locally by the equations

ĝ_{j}(x, x′, x′′) = g_{j}(x) + x′ ^{2} + x′′ ^{2}, (x, x′, x′′) ∈ (ℂ^{n} × ℂ × ℂ, 0).

(cf. [5, p. 223], make the change of variables 2ix′ = x_{0} − t + x_{0}^{d−1}, 2x′′ = x_{0} + t − x_{0}^{d−1}). If F̂_{j} denotes the Milnor fiber of ĝ_{j} then by Sebastiani-Thom (cf. [18]) one has H^{n+1}(F̂_{j})_{1} = H^{n−1}(F_{j})_{1}(−1). Let V_{1} : H^{n−1}(F_{j})_{1} → H_{c}^{n−1}(F_{j})_{1}(−1) be the infinitesimal variation map of type (−1, −1) (cf. [23, (2.5)]), let k : H_{c}^{n−1}(F_{j})_{1} ⟶ H^{n−1}(F_{j})_{1} be the natural map, M_{j,c} : H_{c}^{n−1}(F_{j}) ⟶ H_{c}^{n−1}(F_{j}) the monodromy at the level of cohomology with compact supports. Define N_{1} = log(M_{1}), N_{j,1} = log(M_{j,1}), and N_{j,c,1} = log(M_{j,c,1}). Then from [23] one has:

H_{{p_{j}}}^{n−1}(X^{∞}) = Ker (H_{c}^{n−1}(F_{j})_{1} ⟶^{k} H^{n−1}(F_{j})_{1})

and similarly for X′, so one has

(3.9)  H_{{p′_{j}}}^{n+1} (X′) = H_{{p_{j}}}^{n−1} (X^{∞})(−1).

Since N_{j,c,1} = V_{1}∘k and V_{1} is an isomorphism, one obtains that H_{{p_{j}}}^{n−1}(X^{∞})(−1) = Ker k(−1) = Ker N_{j,c,1}(−1) ≃ Ker N_{j,1}. The last isomorphism is given by V_{1} and follows from the fact that (M_{j,c})_{1} ∘ V_{1} = V_{1} ∘ (M_{j})_{1}.

The above identities, together with (3.9) and (3.8) give:

Ker (N_{1} | P^{n}(X_{t})_{1}) ≃ ⨁_{j=1}^{k} Ker (N_{j,1} | H^{n−1}(F_{j})_{1}).

Now, (3.3) implies that

(3.10)  P^{n}(X_{t})_{1} =^{n} ⊕_{j=1}^{k} H^{n−1}(F_{j})_{1}

(i.e. they have the same Hodge numbers).

Now consider (E.0) and (2.3). The map N_{1}^{0} : H^{n}(X_{t}^{0})_{1} ⟶ H^{n}(X_{t}^{0})_{1} is a morphism of type (−1, −1) and by (2.3):

(3.11)  Im (N_{1}^{0}) = Im [P^{n}(X_{t})_{1} → H^{n}(X_{t}^{0})_{1}].

Via (3.4) and (3.10), identity (3.7) reads

(3.12) \begin{matrix}dim Gr_{n+1+ℓ}^{W} P_{X^{∞}}^{n+1} (X_{t}) + dim Gr_{n+1+ℓ}^{W} P^{n}(X_{t})_{1} \\ = dim Gr_{n+1−ℓ}^{W} P^{n}(X_{t})_{1} − dim Gr_{n+1−ℓ}^{W} P_{X^{∞}}^{n} (X_{t}). \end{matrix}

Using (E.0), this equality can be rewritten as

dim Gr_{n+1+ℓ}^{W} H^{n}(X_{t}^{0})_{1} = dim Gr_{n+1−ℓ}^{W} H^{n}(X_{t}^{0})_{1}.

Together with (3.11), this implies that the weight filtration of H^{n}(X_{t}^{0})_{1} is the monodromy weight filtration centered at n + 1 and its primitive spaces can be identified with P_{X^{∞}}^{n+1}(X_{t}), i.e. for p + q ≥ n + 1

p_{1}^{p,q}(f) = h^{p,q}(P_{X^{∞}}^{n+1} (X_{t})) = h^{n−p,n−q}(P^{n−1}(X^{∞})) = h^{n−q,n−p} (P^{n−1}(X^{∞})).

Case ξ^{d} = 1, ξ ≠ 1.

From (E.0) H^{n}(X_{t}^{0})_{ξ} = P^{n}(X_{t})_{ξ} and from (E.2)

Ker (M − ξ Id | P^{n}(X_{t}) ) = P^{n}(X′_{0})_{ξ}.

Since the weight filtration of P^{n}(X_{t})_{ξ} is centered at n (cf. [19]),

p_{ξ}^{p,q} (f) = h^{n−q, n−p} (P^{n}(X′_{0})_{ξ}).

Case ξ^{d} ≠ 1.

Given ξ with ξ^{d} ≠ 1, from (E.0) we have H^{n}(X_{t}^{0})_{ξ} = P^{n}(X_{t})_{ξ} as MHS’s, and from (E.1’) we have

(3.13)  P^{n}(X_{t})_{ξ} = (⊕_{j} H^{n−1}(F_{j})^{⊕d−1}, ⊕c_{d−1}(M_{j}))_{ξ^{−1}}

as vector spaces. Recall that, as vector spaces with automorphism we have

(3.14)  (H^{n}(F′_{j}), M′_{j}) ≃ (H^{n−1}(F_{j})^{⊕d−1}, c_{d−1}(M_{j})^{d}).

By (E.1), for η ≠ 1 we have an isomorphism of MHS’s:

(3.15)  ⊕_{ξ^{d}=η} P^{n}(X_{t})_{ξ} = ⊕_{j} (H^{n}(F′_{j}), M′_{j})_{η^{−1}}

where (H^{n}(F′_{j}), M′_{j}) is given by the local smoothing π′ : (X′, p′_{j})⟶(D′, 0). In some local coordinates this can be identified with π′ : (Y_{j}, 0) ⟶ (D′, 0), where

Y_{j} = {(y, y_{0}, t) | g_{j}(y) + t y_{0} = y_{0}^{d}}

and π′(y, y_{0}, t) = t (see [5]). We will compute the MHS of the vanishing cohomology of π′ corresponding to eigenvalues ≠ 1. This computation, and the above isomorphisms will provide the result.

The idea of the following construction comes from a recent paper of H. Hamm [8]. We thank J. Steenbrink for drawing our attention to that paper.

Fix an integer m ≥ 1 such that for any eigenvalue λ of M_{j} one has λ^{m} = 1. Then, by (3.14), for any eigenvalue λ of M′_{j} one has λ^{m(d−1)} = 1.

Let (Ỹ_{j}, 0) be the fibre product of π′ : (Y_{j}, 0) → (D′, 0) and (D′′, 0) → (D′, 0) where the last arrow is s ↦ s^{m(d−1)} = t and D′′ is again a small disk. So, (Ỹ_{j}, 0) is locally given by the equation g_{j}(y) + s^{m(d−1)}y_{0} = y_{0}^{d}.

Let π′′ : (Ỹ_{j}, 0) → (D′′, 0) be the projection (y_{0}, y, s) ↦ s. Then the fiber of π′′ is the same as the fiber of π′, its monodromy is the m(d − 1)-th power of the monodromy of π′, in particular it is unipotent. Let K̃ be the real link of (Ỹ_{j}, 0). On K̃ we define the natural Galois action induced by (y_{0}, y, s) ↦ (y_{0}, y, s e(\frac{1}{m(d−1)})).

We denote by V′_{j} : H^{n}(F′_{j}) ⟶ H_{c}^{n}(F′_{j})(−1) the infinitesimal variation map ([23, (2.5)]). By [loc. cit., 2.6.b] one has the following exact sequence of MHS’s:

0 → H^{n}(K̃) → H^{n}(F′_{j}) →^{V′_{j}} H_{c}^{n}(F′_{j})(−1) → H^{n+1}(K̃) → 0 .

(3.16) Lemma: This exact sequence is equivariant with respect to an action which at the level of H^{n}(F′_{j}) is the monodromy action M′_{j} of π′ and on H^{n}(K̃) is the Galois action. Proof of the lemma: If T′_{j,geom} denotes the geometric monodromy of π′, then its lifting composed with the Galois action is isotopic to the identity (by a similar argument as in the proof of E.1’ and E.2 in [6]). Therefore, the inverse of (T′_{j,geom})^{∗} corresponds to the Galois action.

If H^{n}(K̃)_{η} denotes the η-eigenspace (η ≠ 1) with respect to this Galois action, then by (3.16) H^{n}(K̃)_{η} = Ker (V′_{j})_{η}. Recall that k ∘ V′_{j} = N′_{j} where N′_{j} = \frac{1}{m(d−1)} log (M′_{j})^{m(d−1)} and k is the natural map H_{c}^{n}(F′_{j}) ⟶ H^{n}(F′_{j}). Since k_{η} is an isomorphism for η ≠ 1, Ker (N′_{j})_{η} = Ker (V′_{j})_{η}. Therefore, for η ≠ 1,

(3.17)  H^{n}(K̃)_{η} = Ker ((N′_{j})_{η} | H^{n}(F′_{j})_{η}).

The advantage of this identification is the following: The map π′ is defined on a hypersurface singularity germ, and there are very few methods to compute the MHS of its vanishing cohomology. On the other hand, K̃ can be represented as the link of a hypersurface singularity, and there are good methods to compute its MHS. Indeed, let u : (ℂ^{n+2}, 0) ⟶ (ℂ, 0) be defined by u (y, y_{0}, s) = g_{j}(y) + s^{m(d−1)} y_{0} − y_{0}^{d}. Let U be its Milnor fiber. Then by [23] we have the exact sequence of MHS’s

(3.18)  0 → H^{n}(K̃) → H_{c}^{n+1}(U)_{1} →^{k_{1}} H^{n+1}(U)_{1} → H^{n+1}(K̃) → 0 .

Besides the monodromy action, on this exact sequence there is a Galois action induced by the transformation s ↦ s · e(\frac{1}{m(d−1)}), which commutes with the monodromy action.

Let H_{c}^{n+1}(U)_{1,η} be the generalized eigenspace corresponding to eigenvalue 1 with respect to the monodromy action and eigenvalue η with respect to the Galois action. Then (3.18) gives

H^{n}(K̃)_{η} = Ker (k_{1,η} | H_{c}^{n+1}(U)_{1,η}).

If V_{1} : H^{n+1}(U)_{1} ⟶ H_{c}^{n+1}(U)_{1}(−1) is the infinitesimal variation map of u, then V_{1} ∘ k = N_{c,1} (N_{c} being the logarithm of monodromy acting on cohomology with compact supports). Since V_{1} is an isomorphism and N_{c,1} ∘ V_{1} = V_{1} ∘ N_{1}, we get

Ker k_{1,η} = Ker (N_{c,1,η} | H_{c}^{n+1}(U)_{1,η})
= Ker (N_{1,η} | H^{n+1}(U)_{1,η})(+1).

Therefore, for η ≠ 1:

(3.19)  H^{n}(K̃)_{η} = Ker (N_{1,η} | H^{n+1}(U)_{1,η})(+1).

Now, (3.17), (3.19) and (3.3) give the numerical equivalence of H^{n}(F′_{j})_{η} and H^{n+1}(U)_{1,η}(+1), i.e.

(3.20)  H^{pq}(F′_{j})_{η} =^{n} H^{p+1,q+1} (U)_{1,η}.

Let w : (ℂ^{2}, 0) ⟶ (ℂ, 0) be given by w(y_{0}, s) = s^{m(d−1)}y_{0} − y_{0}^{d}. Since w defines an isolated quasihomogeneous singularity, its equivariant Hodge numbers can be computed using [21] (see also [1]). Then, the Sebastiani-Thom result of [18] allows to express the equivariant Hodge numbers of u = g_{j} + w in terms of those of g_{j}.

(3.21) Proposition:

(a) For each 1-eigenvector of (H^{n−1}(F_{j}), M_{j}) of type (p, q) there are d − 2 eigenvectors of (H^{n}(F′_{j}), M′_{j})_{≠1} of type (p, q) and of eigenvalues e(\frac{k}{d−1}) with 1 ≤ k ≤ d − 2.

(b) For each e(i/d)-eigenvector of type (p, q) of (H^{n−1}(F_{j}), M_{j}) with i ∈ ℤ, 0 < i < d, there are d − 2 eigenvectors {v_{k}}_{k} of (H^{n}(F′_{j}), M′_{j})_{≠1} where

k = 0, …, { \atop d − 1 − i}, … d − 2, v_{k} has eigenvalue e(\frac{k+i}{d−1}) and type

(p + [\frac{k + i}{d − 1}] , q + 1 − [\frac{k + i}{d − 1}]).

(c) For each e(γ)-eigenvector of (H^{n−1}(F_{j}), M_{j}) with dγ ∉ ℤ and γ ∈ (0, 1), and type (p, q), there are d − 1 eigenvectors {w_{k}}_{k} of (H^{n}(F′_{j}), M′_{j})_{≠1} where k = 0, …, d − 2 , w_{k} has eigenvalue e(γ + \frac{k+γ}{d−1} ) and type

(p + [γ + \frac{k + γ}{d − 1} ], q + 1 − [γ + \frac{k + γ}{d − 1} ]).

(d) All eigenvectors of (H^{n−1}(F′_{j}), M_{j}) corresponding to eigenvalues ≠ 1 are obtained from a), b) or c) above.

Proof: In order to have a simpler form of Sebastiani-Thom, we will work with spectral pairs. A λ-eigenvector v of type (p, q) of (H^{n−1}(F_{j}), M_{j}) gives a pair S_{v} = (α, p + q − s) where λ = e(−α), n − 1 + [−α] = p and s = 0 if α ∉ ℤ and s = 1 if α ∈ ℤ (cf. §6).

One has Spp(w) = ∑_{(i,j)}(\frac{i}{d} + \frac{j}{md} − 1, 1), where the residue classes of the monomials {y_{0}^{i−1}s^{j−1}} form a base in the Milnor algebra ℂ[y_{0}, s]/(∂w). A possible choice of the monomials is the following: (i, j) either satisfies 1 ≤ i ≤ d and 1 ≤ j ≤ m(d − 1) − 1, or (i, j) = (1, m(d − 1)). Then v contributes to Spp(u) with pairs

∑_{(i,j)} (α + \frac{i}{d} + \frac{j}{md} , p + q − s + 2).

This gives eigenvectors of H^{n+1}(U)_{1,≠1} if and only if

(3.22)  α + \frac{i}{d} + \frac{j}{md} ∈ ℤ and \frac{j}{m(d − 1)} ∉ ℤ.

One has to search for solutions of (3.22) with the restrictions

(3.23)  1 ≤ i ≤ d , 1 ≤ j ≤ m(d − 1) − 1.

In case (a) (resp. (b), resp. (c)) one has S_{v} = (n − p − 1, p + q − 1) ( resp. S_{v} = (−i/d − p + n − 1, p + q), resp. S_{v} = (−γ − p + n − 1, p + q) ). One gets (a), (b), (c) by solving (3.22), (3.23) in each case. The details of the computation are left to the reader.

Now we can finish the proof of theorem 3.1 (case ξ^{d} ≠ 1). Recall the identities (3.13-14-15). Subcase 1: Assume ξ^{d−1} = 1, i.e. ξ = e(−\frac{ℓ}{d−1}) with 1 ≤ ℓ ≤ d − 2. Since ξ^{−1} is an eigenvalue of c_{d−1}(M_{j}) (cf. 3.13), ξ^{1−d} = 1 is an eigenvalue of M_{j}. By (3.21.a), each 1-eigenvector of M_{j} of type (p, q) gives d − 2 eigenvectors of M′_{j} of type (p, q), with eigenvalues η^{−1} = e(\frac{k}{d−1}) (1 ≤ k ≤ d − 2). But (in (3.15)) only one corresponds to ξ (i.e. satisfies e(−\frac{k}{d−1}) = ξ^{d}), namely that with k = ℓ. So, each 1-eigenvector of M_{j} of type (p, q) gives one eigenvector of (H^{n}(X_{t}^{0}), M_{f}^{∞})_{ξ} of type (p, q). Therefore,

p_{ξ}^{p,q}(f) = ∑_{j} p_{1}^{p,q}(X^{∞}, p_{j}).

Subcase 2: Assume ξ^{(d−1)d} = 1 but ξ^{d−1} ≠ 1. Then ξ = e(−β), where β = \frac{ℓ}{d−1} + \frac{i}{d(d−1)}, with 0 < i < d, 0 ≤ ℓ ≤ d − 2. Then α = ξ^{1−d} = e(\frac{i}{d}) is an eigenvalue of M_{j} with α ≠ 1, α^{d} = 1. Using (3.21.b) one gets eigenvalues η^{−1} = e(\frac{k+i}{d−1}) for M′_{j}, but only one corresponds to ξ (because e(−\frac{k+i}{d−1}) = ξ^{d} implies k = ℓ). Thus, from (3.21.b), (3.13) and (3.15) one has:

p_{ξ}^{p,q}(f) = ∑_{j} p_{ξ^{1−d}}^{p−[ \frac{ℓ+i}{d−1} ], q−1+[ \frac{ℓ+i}{d−1} ]}(X^{∞}, p_{j}).

Now, notice that γ := \{(d − 1)β\} = \frac{i}{d}, hence [β + γ] = [\frac{ℓ+i}{d−1}].

Subcase 3: ξ^{d(d−1)} ≠ 1. Follows from (3.21.c) as in the subcases above (now k = [(d − 1)β]).

(3.24) Remark : The proof of theorem (3.1) in case ξ = 1 suggests the existence of a natural perfect pairing

D : P^{n−1}(X_{s}^{∞})_{1} ⊗ H^{n}(X_{t}^{0})_{1} ⟶ ℚ(−n).

Our proof gives this duality at the level of Hodge numbers and we expect that there is a geometrical construction of D. We expect that H^{n}(X_{t}^{0})_{1} is not pure if V ar is not an isomorphism (cf. (3.2.b) ). This suggests that D should be constructed from the Milnor fibration at infinity (Notice that the MHS on P^{n−1}(X_{s}^{∞})_{1} depends on the choice of the smoothing {X_{s}}_{s} of X^{∞}; also the MHS of H^{n}(X_{t}^{0})_{1} depends on the lower order terms f_{d−1} + f_{d−2} + ⋯ of f. But maybe a good pairing of these parameter spaces gives rise to a duality D).

4. The Hodge numbers of X^{∞} and X′_{0}.

Theorem (3.1) gives the MHS of ⊕_{ξ^{d}=1} H^{n}(X_{t}^{0})_{ξ} in terms of the MHS of X^{∞} and X′_{0}. In the sequel we would like to separate the local and the global information. We say that an invariant is local if it depends only on the type of the hypersurface singularities {(X^{∞}, p_{j})}_{j=1}^{k} and some universal constant depending on n and d. If an invariant depends on the positions of the singularities of X^{∞}, then we say that it is global. For example, dim P^{n}(X^{∞}) and dim P^{n−1}(X^{∞}) are global, but their difference is local.

(4.1) We start defining some universal constants: If Q ∈ ℂ [z_{1}, …, z_{n+1}] is an homogeneous polynomial of degree d which defines a smooth hypersurface V = V (Q) ⊆ ℙ^{n}, then the Hodge numbers of V can be computed as follows (cf. [7]): Let (∂Q) denote the ideal in ℂ [z_{1}, …, z_{n+1}] generated by the partial derivatives of Q and set F(Q) = ℂ [z_{1}, …, z_{n+1}]/(∂Q). Let F(Q)_{ℓ} be the subspace generated by the monomials of degree ℓ, and set

Ω_{n} = ∑_{i=1}^{n+1} (−1)^{i} z_{i} dz_{1} ∧ … ∧ d̂z_{i} ∧ … ∧ dz_{n+1}.

If P is an homogeneous polynomial of degree deg P = (k + 1) d − n − 1 for some k ≥ 0, then P ·Ω_{n} / Q^{k+1} defines a rational differential on ℙ^{n} with poles along the hypersurface V. Its residue class Res(P Ω_{n} / Q^{k+1}) lives in the primitive cohomology P^{n−1} (V) and one has an isomorphism (cf. [7, (4.6)])

(4.2)  F (Q)_{(k+1) d−n−1} ⟶^{∼} P^{n−1−k, k} (V)

given by P ⟼ Res(P Ω_{n} / Q^{k+1}).

It is well known that the Hodge numbers of the cohomology of V (equivalently, dim F (Q)_{ℓ}) does not depend on the choice of Q as long as V (Q) ⊆ ℙ^{n} is smooth. We denote

(4.3)  j_{k,s}^{n,d} = dim F (Q)_{(k+1) d−(n+1)−s}

where 0 ≤ s < d and k ≥ 0. The next theorem gives the Hodge numbers of P^{n−1}(X^{∞}) in terms of the Hodge numbers of P^{n}(X^{∞}) modulo local contributions. After this paper was written, we have got to know of a very recent paper of A. Dimca ([4]) where the same theorem is proved, under a slightly different form.

(4.4) Theorem: Let X^{∞}, {p_{1}, …, p_{k}}, F_{1}, …, F_{k} be as in the previous sections. Then:

(a) h^{i, n−k−i} (P^{n−1} (X^{∞}) ) = ∑_{j} p_{1}^{k+i−1, n−1−i} (X^{∞}, p_{j}) if k ≥ 3.

(b) h^{i, n−2−i} (P^{n−1} (X^{∞}) ) = ∑_{j} p_{1}^{i+1, n−1−i} (X^{∞}, p_{j})−h^{i+1, n−1−i} (P^{n}(X^{∞}) ).

(c) h^{i, n−1−i} (P^{n−1} (X^{∞}) ) = j_{i,0}^{n,d} − ∑_{j} dim Gr_{F}^{i} H^{n−1} (X^{∞}, p_{j})

−∑_{j} ∑_{q>i} p_{1}^{q, n−1−i} (X^{∞}, p_{j})+h^{i, n−i} (P^{n}(X^{∞}) )+h^{i+1, n−1−i} (P^{n} (X^{∞}) ).

Proof: (a) and (b) follow from (3.6) or (3.7). For (c), consider the following exact sequence (similarly as in (3.6), but now for all eigenvalues):

(4.5) 0 → P^{n−1} (X^{∞}) → P^{n−1}(X_{s}^{∞}) → ⊕_{j} H^{n−1} (F_{j}) → P^{n} (X^{∞}) → 0.

Passing to the limit MHS does not change the dimension of the terms of the Hodge filtration, thus dim Gr_{F}^{i} P^{n−1} (X_{s}^{∞}) = j_{n−1−i,0}^{n,d} = j_{i,0}^{n,d} (cf. 4.2). Now take Gr_{F}^{i} on the sequence above and apply (a) and (b).

(4.6) Corollary: If p + q ≥ n + 3, then p_{1}^{p,q} (f) is local. Proof: Use the theorem above and theorem (3.1,a).

Let X′_{0} = {f_{d}(z_{1}, …, z_{n+1}) + z_{n+2}^{d} = 0} ⊆ ℙ^{n+1} be the d-th fold cyclic covering of ℙ^{n} branched along X^{∞}. The Galois action of ℤ/dℤ is given by ξ ∗ [z_{1} : …: z_{n+2}] = [z_{1} : …: ξ z_{n+2}] (ξ^{d} = 1). Recall that P^{∗}(X′_{0})_{ξ} denotes the ξ-eigenspace of P^{∗}(X′_{0}) with respect to the Galois action.

The singularities Sing X′_{0} = {p′_{1}, …, p′_{k}} are the d-th suspensions of the singularities of X^{∞}. If (X^{∞}, p_{j}) is locally given in ℂ^{n} by g_{j}(x_{2}, …, x_{n+1}) = 0, then (X′_{0}, p′_{j}) is given by g̃_{j} := g_{j}(x_{2}, …, x_{n+1}) + x_{n+2}^{d} = 0 in ℂ^{n+1}. Let F̃_{j} be the Milnor fiber of g̃_{j}, M̃_{j} its monodromy acting on H^{n}(F̃_{j}). Then on H^{n}(F̃_{j}) there is a Galois action (given by ξ∗(x_{2}, …, x_{n+2}) = (x_{2}, …, ξ x_{n+2}) if ξ^{d} = 1) and this action commutes with M̃_{j}. H^{n}(F̃_{j}) carries a MHS, we consider the space of primitive elements given by the nilpotent part of M̃_{j}. Let P_{1}^{ab} (X′_{0}, p′_{j}) be the space of primitive vectors of type (a, b) corresponding to the eigenvalue 1 of M̃_{j}. Finally, let p_{1}^{ab} (X′_{0}, p′_{j})_{ξ} be the dimension of the ξ-eigenspace of the induced Galois action on P_{1}^{a,b} (X′_{0}, p′_{j}). (In order to eliminate any confusion, we emphasize that below in (4.7.c) H^{n}(F̃_{j})_{ξ} is the ξ–eigenspace with respect to the Galois action).

(4.7) Theorem: Let ξ = e(s/d), 0 < s < d. Then with the above notations, one has

(a) h^{i, n+1−k−i} (P^{n}(X′_{0})_{ξ}) = ∑_{j} p_{1}^{k+i−1, n−i} (X′_{0}, p′_{j})_{ξ} if k ≥ 3

(b) h^{i, n−1−i} (P^{n}(X′_{0})_{ξ}) = ∑_{j} p_{1}^{i+1, n−i} (X′_{0}, p′_{j})_{ξ} − h^{i+1, n−i} (P^{n+1}(X′_{0})_{ξ}).

(c) h^{i, n−i} (P^{n}(X′_{0})_{ξ}) = j_{n−i,s}^{n,d}−∑_{j} dim Gr_{F}^{i} H^{n}(F̃_{j})_{ξ}−∑_{j} ∑_{q>i} p_{1}^{q, n−i} (X′_{0}, p_{j})_{ξ} + h^{i, n+1−i} (P^{n+1}(X′_{0})_{ξ}) + h^{i+1, n−i} (P^{n+1}(X′_{0})_{ξ}).

Proof: A deformation (f_{d})_{s} of f_{d} induces a deformation (f_{d})_{s} + z_{n+2}^{d} of f_{d} + z_{n+2}^{d}. Let V ( (f_{d})_{s} + z_{n+2}^{d}) = X′_{s} ⊆ ℙ^{n+1}. Then the exact sequence of vanishing cycles is

(4.8)  0 → P^{n}(X′_{0}) → P^{n}(X′_{s}) → ⨁_{j} H^{n}(F̃_{j}) → P^{n+1}(X′_{0}) → 0.

On this sequence there are a monodromy and a Galois action which commute. Now the proof of (4.7) is the same as the proof of (4.4) if we replace the exact sequence (4.5) by (4.8) (and we take the ξ-eigenspaces, with respect to the Galois action). The remaining part is the computation of dim Gr_{F}^{i} P^{n}(X′_{s})_{ξ} (for s ≠ 0 fixed). Take i = n − k. By the discussion in (4.1) and with the same notations, H^{n−k, k} (P^{n}( V (Q + z_{n+2}^{d}) )_{ξ}) is given by the residues of rational differentials of type P̃ Ω_{n+1} / (Q + z_{n+2}^{d})^{k+1} where deg P̃ + (n + 2) = (k + 1) d. Since Ω_{n+1} is homogeneous of degree one in z_{n+2}, such a form lives in the ξ-eigenspace with respect to the Galois action if and only if P̃ = P (z_{1}, …, z_{n+1})z_{n+2}^{s−1}, hence P ∈ F(Q)_{(k+1)d−(n+1)−s}.

(4.9) Corollary: If p + q ≥ n + 2, then p_{ξ}^{p,q} (f) is local for any ξ^{d} = 1, ξ ≠ 1.

In the next section we will study the polarization properties of H^{n} (X_{t}^{0}). This is not difficult for the generalized eigenspace H^{n} (X_{t}^{0})_{≠1}, but for H^{n}(X_{t}^{0})_{1} we need a suplementary construction.

Notice first that if f = f_{d}+… is a (∗)-polynomial, then f̃ = f (x_{1}, …, x_{n+1})+ x_{n+2}^{d} : ℂ^{n+2} → ℂ is again a (∗)-polynomial. We will compare the equivariant Hodge numbers and the polarization properties of f for eigenvalue 1 and of f̃ corresponding to eigenvalues ≠ 1. In order to do this, we need a Sebastiani-Thom-type property for the MHS at infinity of f̃ = f + x_{n+2}^{d} (at least for eigenvalues ξ with ξ^{d} = 1). This will be established now. Let X′′_{0} = {f_{d} (z_{1}, …, z_{n+1}) + z_{n+2}^{d} + z_{n+3}^{d} = 0} ⊆ ℙ^{n+2} be the d-fold cyclic covering of ℙ^{n+1} branched along X′_{0}. The Galois action on X′′_{0} is given by multiplication by e(1/d) on the last coordinate z_{n+3}. Then one has

(4.10) Theorem: For ξ = e(s/d), 0 < s < d and k ∈ {n, n − 1},

\begin{matrix}h^{p+1, q+1} (P^{k+2} (X′′_{0})_{ξ}) = h^{p,q} (P^{k} (X^{∞}) ) \\ + ∑_{{0<t<d \atop t+s≠d}} h^{p+[ \frac{s+t}{d}], q+1−[ \frac{s+t}{d}]} (P^{k+1} (X′_{0})_{e( \frac{s+t}{d})}). \end{matrix}

Proof: Similarly as in the proof of (4.7), consider a deformation (f_{d})_{s} of f_{d}, which gives deformations (f_{d})_{s} + z_{n+2}^{d}, (f_{d})_{s} + z_{n+2}^{d} + z_{n+3}^{d}. We consider the exact sequences (4.5), (4.8) and the analogous one for (f_{d})_{s} +z_{n+2}^{d} +z_{n+3}^{d} and we compare dim Gr_{F}^{p} of the corresponding terms. First we concentrate our attention on the second terms in these exact sequences: P^{n−1} (X_{s}^{∞}), P^{n} (X′_{s}), P^{n+1} (X′′_{s}). Since dim Gr_{F}^{p} remains constant when we pass to a limit MHS, in order to compute it we can replace this MHS by the pure Hodge structures given by a fixed s_{0} (s_{0} ≠ 0). Set Q = (f_{d})_{s_{0}}, X = {Q = 0} ⊆ ℙ^{n}, X′ = {Q + z_{n+2}^{d} = 0} ⊆ ℙ^{n+1}, X′′ = {Q + z_{n+2}^{d} + z_{n+3}^{d}} ⊆ ℙ^{n+2}. Notice that there is also an action of ℤ/dℤ on X′ (resp. X′′) given by multiplication of z_{n+2} (resp. z_{n+3}) by e(1/d). Again by Griffiths’ construction, P^{n+1−(q+1), q+1} (X′′) is freely generated by the classes P Ω_{n+2} (Q + z_{n+2}^{d} + z_{n+3}^{d})^{−(q+2)} with deg P + n + 3 = d (q + 2), P ∈ F (Q + z_{n+2}^{d} + z_{n+3}^{d}). Moreover, the ξ-eigenspace is given by the classes corresponding to polynomials of the form P = z_{n+3}^{s−1} P̃ (z_{1}, …, z_{n+2}). Write P̃ = ∑_{0<t<d} z_{n+2}^{t−1} P_{t−1}(z_{1}, …, z_{n+1}). Then deg P_{t−1} + t − 1 + s − 1 + n + 3 = d (q + 2), and

(4.11)  P^{n+1−(q+1), q+1} (X′′)_{ξ} = ∑_{0<t<d} F(Q)_{d (q+2)−n−1−t−s}.

If t + s < d then it follows that deg P_{t−1} + t + s − 1 + n + 2 = d (q + 2), hence these polynomials P_{t−1} provide rational differentials z_{n+2}^{t+s−1} P_{t−1} Ω_{n+1} (Q + z_{n+2}^{d})^{−(q+2)} generating P^{n−(q+1), q+1} (X′)_{e( \frac{s+t}{d})}. If t + s > d, then

deg P_{t−1} + t + s − 1 − d + n + 2 = d (q + 1)

hence these P_{t−1}’s provide forms z_{n+2}^{t+s−1−d} P_{t−1} Ω_{n+1}(Q + z_{n+2}^{d})^{−(q+1)} generating P^{n−q, q} (X′)_{e( \frac{s+t}{d})}. Finally, if t + s = d then deg P_{t−1} + n + 1 = d (q + 1) hence P_{t−1} Ω_{n} Q^{−(q+1)} are rational forms on ℙ^{n} which generate P^{n−1−q, q} (X) via Griffith’s isomorphism. Therefore

(4.12) p^{p+1, q+1}(X′′)_{e(s/d)} = p^{p,q}(X) + ∑_{{0<t<d \atop t+s≠d}} p^{p+[ \frac{t+s}{d}], q+1−[ \frac{t+s}{d}]}(X′)_{e( \frac{t+s}{d})}.

Passing to the limit we obtain

(4.13)  dim Gr_{F}^{p+1} P^{n+1}(X′′_{s})_{ξ} = dim Gr_{F}^{p} P^{n−1} (X_{s}^{∞})

+ ∑_{{0<t<d \atop t+s≠d}} dim Gr_{F}^{p+[ \frac{t+s}{d}]} P^{n}(X′_{s})_{e( \frac{t+s}{d})}.

Now we consider the third (the local) terms in the sequences (4.5), (4.8) and the one for (f_{d})_{s} + z_{n+2}^{d} + z_{n+3}^{d}. There is a local analogue of the above global argument where the rational differential forms are replaced by forms in the Brieskorn lattice.

(4.14) Lemma: Consider an isolated singularity given by a map germ g : (ℂ^{n}, 0) → (ℂ, 0) (with local coordinates (x_{2}, …, x_{n+1}) ), set g′ = g+x_{n+2}^{d}, g′′ = g + x_{n+2}^{d} + x_{n+3}^{d}. The Galois group ℤ/dℤ acts on the vanishing cohomology corresponding to g′ (resp. g′′), at the level of coordinates this action is given by multiplication of x_{n+2} (resp. x_{n+3}) by e(1/d). Let H_{λ}^{p}(g′)_{ξ} be the (λ, ξ)- generalized eigenspace of the p-th vanishing cohomology group of g′ where ξ (resp. λ) is an eigenvalue of the Galois action (resp. of the monodromy), and similarly for g′′. This actions are compatible with the MHS on the vanishing cohomology, let p_{λ}^{pq}(g′)_{ξ} be the dimension of the space of primitive elements of type (p, q). Then for ξ = e(s/d) (0 < s < d) and any λ one has

p_{λ}^{p+1, q+1} (g′′)_{ξ} = ∑_{{0<t<d \atop t+s≠d}} p_{λ}^{p+[ \frac{t+s}{d}], q+1−[ \frac{t+s}{d}]} (g′)_{e( \frac{t+s}{d})} + p_{λ}^{pq} (g).

Proof: Since the information provided by the equivariant primitive cohomology Hodge numbers is the same as the one in the set of spectral pairs, we will give the proof at the level of spectral pairs. If (H, T) is a MHS with a semisimple finite action, we refer to [22] (5.3) or [10] for the definition of the set of spectral pairs Spp (H, T) (or see section 6). By the Sebastiani-Thom formula proved in [18] (which obviously is compatible with the Galois action), we have:

(4.15) Spp H_{λ}^{n+1} (g′′)_{ξ} = ∑_{β+γ≡α(mod ℤ)} S_{β}(g) ∗ Spp H′_{e(−γ)} (x_{n+2}^{d} + x_{n+3}^{d})_{ξ}

where λ = e(−α), S_{β}(g) = Spp H_{e(−β)}^{n−1} (g) and (α, ω) ∗ (α′, ω′) = (α + α′ + 1, ω + ω′ + 1). Since h = x_{n+2}^{d} +x_{n+3}^{d} is homogeneous, its vanishing cohomology is generated (in the Brieskorn lattice) by the forms ω = x_{n+2}^{u−1} x_{n+3}^{v−1} (h − 1)^{[− \frac{u+v}{d}]} d x_{n+2} ∧ d x_{n+3}, where 0 < u, v < d. The Galois action on this form is e(1/d) ∗ ω = e(v/d)ω and the spectral pair provided by ω is (\frac{u+v}{d} − 1, 1). Therefore

Spp H_{e(−γ)}^{1} (h)_{ξ} = \{(\frac{u + s}{d} − 1, 1)\}

where γ = \frac{u+s}{d} − 1. So (4.15) reads:

(4.16)  Spp H_{λ}^{n+1} (g′′)_{ξ} = ∑ S_{β} (g) ∗ (\frac{t + s}{d} − 1, 1)

where the sum is over 0 < t < d, β + \frac{t+s}{d} − α ∈ ℤ.

If we denote T (a, b) the transformation on ℤ[ℚ × ℤ] given by T (a, b) (α, ω) = (α + a, ω + a + b), then (again by the Sebastiani–Thom theorem), we have

S_{β}(g) ∗ (\frac{t + s}{d} − 1, 1) = \{ \begin{matrix}T ([\frac{t+s}{d}], 1 − [\frac{t+s}{d}]) Spp H_{λ}^{n}(g′)_{e( \frac{s+t}{d})} & if t + s ≠ d \\ T (1, 1) Spp H_{λ}^{n−1}(g) & if t + s = d. \end{matrix}

This finishes the proof of (4.14). We return now to the proof of (4.10). Consider the middle maps in the sequences (4.5), (4.8) and the one for (f_{d})_{s} + z_{n+2}^{d} + z_{n+3}^{d}. Both the sources and the targets of these maps have decompositions compatible with them. Thus their kernels and cokernels also decompose, so for k = n − 1, n one has

(4.17) dim Gr_{F}^{p+1} P^{k+2}(X′′_{0})_{ξ} = ∑_{{0<t<d \atop s+t≠d}} dim Gr_{F}^{p+[ \frac{t+s}{d}]} P^{k+1} (X′_{0})_{e( \frac{s+t}{d})}

+ dim Gr_{F}^{p} P^{k}(X^{∞}).

If k = n, then (4.10) follows since the corresponding Hodge structures are pure. If k = n − 1, then apply (4.7), (4.11), (4.14) and the case k = n.

5. Polarization. Connection with the real Seifert form at infinity.

(5.1) First we recall the classification of simple ε-hermitian variation structures. The basic reference is [10], see also [12]. Let U be the complexification of a finite dimensional real vector space, set U^{∗} = Hom_{ℂ}(U, ℂ). We denote θ : U ⟶ U^{∗∗} the natural isomorphism ( θ(u) (φ) = φ(u) ). A bar over an element of U denotes complex conjugation. If φ ∈ Hom_{ℂ} (U, U′) then we denote φ̄ ∈ Hom_{ℂ} (U, U′) the map φ̄ (x) = \overline{φ (x̄)}. The dual map φ^{∗} : U^{′∗} ⟶ U^{∗} of φ is defined by φ^{∗}(ψ) = ψ ∘ φ.

(5.2) Definition ([10, (2.1)]): An ε-hermitian variation structure over ℂ (ε = ±1), abbreviated in the sequel as ε-HV S, is a system (U, b, h, V) where U is the complexification of a finite dimensional real vector space and

(a) b : U ⟶ U^{∗} is a ℂ-linear morphism with \overline{b^{∗} ∘ θ} = ε b

(b) h is a b-orthogonal automorphism of U, i.e. \overline{h^{∗}} ∘ b ∘ h = b

(c) V : U^{∗} ⟶ U is a ℂ-linear morphism with \overline{θ^{−1} ∘ V^{∗}} = −ε V and V ∘ b = h − Id.

If V is an isomorphism, then (U, b, h, V) is called simple and in this case h = −ε V \overline{(θ^{−1} ∘ V^{∗})}^{−1} and b = −V^{−1} − ε \overline{(θ^{−1} ∘ V^{∗})}^{−1}.

(5.3) Example: Let f : ℂ^{n+1} ⟶ ℂ be a polynomial map which admits a Milnor fibration at infinity (e.g. f is a (∗)-polynomial, cf. (2.1 b) ). Let F be the fiber of the Milnor fibration, set U_{ℝ} = H_{n}(F, ℝ), U_{ℝ}^{∗} = H^{n}(F, ℝ) (which is identified with H_{n}(F, ∂F, ℝ) via the perfect pairing H_{n}(F, ∂F, ℝ) ⊗ H^{n}(F, ℝ) ⟶ ℝ). Let h_{ℝ} : U_{ℝ} ⟶ U_{ℝ} be the real (classical) algebraic monodromy, b_{ℝ} : U_{ℝ} ⟶ U_{ℝ}^{∗} the intersection form and Var : U_{ℝ}^{∗} ⟶ U_{ℝ} the variation map. Then V ar is an isomorphism and (U_{ℝ}, b_{ℝ}, h_{ℝ}, Var) ⊗_{ℝ} ℂ is a (−1)^{n} − HV S which will be denoted V(f). Notice, that similarly as in the local case, V ar is the inverse of the (real) Seifert form of the Milnor fibration at infinity. If f is a (∗)-polynomial, then the Milnor fibration is equivalent to the fibration of f at infinity, hence F is diffeomorphic to the generic fiber of f and (h_{ℝ}^{∗})^{−1} is exactly T_{f}^{∞}.

(5.4) In the sequel, if (U, b, h, V) is a ε-HV S we will assume that the eigenvalues λ of h satisfy |λ| = 1 (for the general case, see [10]). In the next examples, J_{k} denotes the k × k Jordan block with eigenvalue 1.

(a) If b is an isomorphism, then V = (h − Id) b^{−1} is determined from h, b. Up to isomorphism, there are exactly two non-degenerate k×k matrices b such that b̄^{∗} = ε b and J_{k}^{∗} b J_{k} = b. They are determined by the sign of the entry b_{1,k} and will be denoted b_{±}^{k}. If we set ε = (−1)^{n}, we can take (b_{±}^{k})_{1,k} = ±i^{−n^{2}−k+1}. In particular, for λ ≠ 1 ( |λ| = 1) there are exactly two simple (−1)^{n} − HV S’s:

W_{λ}^{k}(±1) = (ℂ^{k}, b_{±}^{k}, λ J_{k}, (λ J_{k} − Id) (b_{±}^{k})^{−1}).

(b) If λ = 1 and k = 1 there are two simple (−1)^{n} − HV S’s

W_{1}^{1} (±1) = (ℂ, 0, Id_{ℂ}, ±i^{n^{2}+1})

(c) If λ = 1 and k ≥ 2, then h = J_{k} and b is not an isomorphism.

Again, there are two simple (−1)^{n} − HV S’s

W_{1}^{k}(±1) = (ℂ^{k}, b̃_{±}^{k}, J_{k}, Ṽ_{±}^{k})

and they are determined by (the sign of) (b̃_{±}^{k})_{k,2} = ±i^{−n^{2}−k+2}. This structure can be recognized also from ( (Ṽ_{±}^{k})^{−1})_{1,k} = ±i^{−n^{2}−k}.

(d) −(U, b, h, V) denotes the structure (U, −b, h, −V) (with the same ε).

The sign conventions are motivated by Hodge-theoretical reasons, they correspond to the sign conventions of the polarisations of Hodge structures. The structure theorem of ε-HV S is the following:

(5.5) Theorem ([10, (2.9)]): Any simple ε-HV S decomposes as a sum of indecomposable ones, the decomposition is unique up to isomorphism and order of summands. If the eigenvalues λ of h satisfy |λ| = 1, then the indecomposable structures are W_{λ}^{k}(±1) where k ≥ 1.

(5.6) Remark: Two real variation structures are isomorphic over ℂ if and only if they are isomorphic as real structures (cf. [10, (2.10)]. This implies that if f is a polynomial which admits a Milnor fibration at infinity the information contained in V (f) (see (5.3) above) is the same as the one contained in the real Seifert form of the Milnor fibration at infinity. If f is a (∗)-polynomial, the connection between V (f) and the Hodge theoretical invariants introduced in sections 1-4 is the following

(5.7) Theorem: Let f be a (∗)-polynomial. Then:

(−1)^{n} · V (f) = ⨁_{λ} ⨁_{2n≥a+b≥n+s} p_{λ}^{b,a} (f) · W_{λ}^{r+1} ( (−1)^{b})

where s = 0 if λ ≠ 1, s = 1 if λ = 1 and r = a + b − n − s ≥ 0. In particular, the equivariant Hodge numbers of H^{n}(X_{t}^{0}) determine completely the real Seifert form of the Milnor fibration at infinity.

Proof: The proof follows closely that of Theorem (6.1) in [10] (which gives the same relation in the case of hypersurface singularities. In [loc.cit.] there is a slightly different definition of the structures W_{λ}^{k}, and in the present theorem we made a sign-correction.) Notice that V (f) has a direct sum decomposition V (f) = ⊕_{λ} V_{λ} (f) given by the eigenvalues of the monodromy h. So, it will be enough to show that one has the identity:

(5.8)  (−1)^{n} · V_{λ} (f) = ⨁_{a,b} p_{λ}^{b,a} (f) W_{λ}^{r+1} ( (−1)^{b}).

In order to see this, first notice that one can define a cohomological variation structure V_{coh}(f) of f by U_{ℝ} = H_{c}^{n}(F, ℝ), U_{ℝ}^{∗} = H^{n}(F, ℝ) (via the perfect pairing H_{c}^{n}(F, ℝ)⊗H^{n}(F, ℝ) ⟶ ℝ); h_{ℝ} = T_{geom}^{∗} and V ([ω]) = [T_{geom}^{∗}(ω)−ω]. It turns out that (−1)^{n} · V(f) = V_{coh}(f).

Assume first that λ ≠ 1. Then by (E.0) we have H^{n}(X_{t}^{0})_{λ} ≃ P^{n}(X_{t})_{λ} as MHS, therefore V_{coh}(f)_{λ} can be determined from the polarization properties of the limit MHS of Schmid. More precisely, if N_{λ}^{0}(T_{f}^{∞}) is the infinitesimal nilpotent monodromy operator associated with T_{f}^{∞}, then the polarization given by N_{λ}^{0}(T_{f}^{∞}) provides:

V_{coh}(f)_{λ} = ⨁_{a,b} p_{λ}^{a,b}(T_{f}^{∞}) W_{λ}^{r+1} ( (−1)^{b}),

where p_{λ}^{a,b}(T_{f}^{∞}) are the equivariant primitive Hodge numbers determined by the classical monodromy T_{f}^{∞}, hence they are equal to pa,b ; λ̄(f) = p_{λ}^{b,a}(f).

Suppose now that λ = 1. Consider the polynomial f + x_{n+2}^{d} : ℂ^{n+2} ⟶ ℂ. As remarked before, it is easy to see that f + x_{n+2}^{d} is again a (∗)-polynomial. We want to compare V (f + x_{n+2}^{d}) with V (f). The ε-HVS of the polynomial map x_{n+2} ⟼ x_{n+2}^{d} is ⊕_{{η^{d}=1 \atop η≠1}} W_{η}^{1}(+1) (with ε = (−1)^{0} = +1). By [11] (which

solves the Sebastiani–Thom problem for Seifert form at infinity associated with “good” polynomials) one has

(5.9) \begin{matrix}V_{ξ}(f + x_{n+2}^{d}) = (−1)^{n+1}[V(f) ⊗ ( ⨁_{{η^{d}=1 \atop η≠1}} W_{η}^{1}(+1))]_{ξ} \\ = (−1)^{n+1}V_{1}(f) ⊗ W_{ξ}^{1}(+1) + (−1)^{n+1} ⨁_{{η≠1 \atop η≠ξ}} V_{η^{−1}ξ}(f) ⊗ W_{η}^{1}(+1). \end{matrix}

One can verify that if η = e(α) and ξ = e(β) with α, β ∈ (0, 1) and α+β ≠ 1, then:

W_{η}^{k}(±1) ⊗ W_{ξ}^{1}(+1) = (−1)^{n} · W_{ηξ}^{k}(±κ(η, ξ)),

where κ(η, ξ) = +1 if α + β < 1 and κ(η, ξ) = −1 if α + β > 1. One has as well:

W_{1}^{k}(±1) ⊗ W_{ξ}^{1}(+1) = (−1)^{n} · W_{ξ}^{k}(±1).

From these formulas and (5.9), it follows that we can determine V_{1}(f) from V_{λ≠1}(f) and V_{λ≠1}(f + x_{n+2}^{d}). But, as showed above, these last ε-HVS can be computed in terms of equivariant Hodge numbers of f and f + x_{n+2}^{d}. These equivariant Hodge numbers are related to those of f for eigenvalue 1 ( using (3.1) and (4.10), the latter applied to k = n − 1), and then formula (5.8) for λ = 1 follows, the details of the computation are left to the reader. This ends the proof.

(5.10) The formula given in this theorem is the exact analogue of the corresponding formula for isolated hypersurface singularities (cf. [10]). Thus, roughly speaking, any connection between Hodge numbers and topological invariants which holds for isolated hypersurface singularities holds also for (∗)-polynomials if we replace the MHS of the local vanishing cohomology by the MHS at infinity, the Milnor fiber by the generic fiber and the local monodromy by the monodromy at infinity. We resume and illustrate this in the next theorem.

(5.11) Theorem: Let f : ℂ^{n+1} ⟶ ℂ be a (∗)-polynomial.

a) H^{n}(X_{f}^{0})_{≠1} and H_{c}^{n}(X_{f}^{0})_{≠1} carry MHS’s which are dual with respect to ℚ(−n), the weight filtration of H^{n}(X_{f}^{0})_{≠1} is the monodromy weight filtration of N_{≠1}^{0} centered at n. This space has a natural (−1)^{n}-symetric polarization form (the Poincaré dual of the intersection form ⟨ · , j · ⟩) and it is polarized by the infinitesimal monodromy −N_{≠1}^{0} (in the sense of Cattani-Kaplan).

b) H^{n}(X_{f}^{0})_{1} and H_{c}^{n}(X_{f}^{0})_{1} carry MHS’s which are dual with respect to ℚ (−n). The weight filtration of H^{n}(X_{f}^{0})_{1} is the monodromy weight filtration of N_{1}^{0} centered at n+1. The infinitesimal variation map ([23, (2.5)] V_{1} : H^{n}(X_{f}^{0})_{1} ⟶ H_{c}^{n}(X_{f}^{0})_{1}(−1) is an isomorphism of MHS’s of type (−1, −1). H^{n}(X_{f}^{0})_{1} has a natural (−1)^{n+1}-symmetric polarization form given by ⟨· , V_{1} · ⟩ and it is polarized by the monodromy −N_{1}^{0}.

Proof: Case a) follows from (E.0) and Schmid’s results [19], which assures that the MHS is polarized by log(T_{f}^{∞})_{u} = −N^{0}. For b), use the polarization properties of W_{1}^{r+1}(± 1) which behaves as in the local case. For the local case see [23] and [21] or [13].

(5.12) Corollary: The equivariant signature σ_{λ}(f) of the generic fiber of f, with respect to the monodromy at infinity, is given by:

σ_{λ}(f) = ∑_{2n≥a+b≥n+s} (−1)^{b}p_{λ}^{a,b} (f) · (1 + (−1)^{a+b−n})/2,

where s = 0 if λ ≠ 1, and s = 1 if λ = 1. The proof is analogous to that of [10, (6.6)], use the formula

σ(W_{λ}^{k}(±1)) = ±\frac{1 + (−1)^{k+1+s}}{2} .

6. The spectral pairs of a (∗)-polynomial.

As in the local case (cf. [22, 18], see also [1, 13.3.A] and [10]) the equivariant Hodge numbers h_{λ}^{p,q} of the MHS at infinity of a polynomial f can be codified in the set of spectral pairs Spp(f) ∈ ℤ[ℚ × ℕ] defined by

Spp(f) = ∑_{(α,ω)} h_{e(−α)}^{n+[−α], ω+s_{α}−n−[−α]} (α, ω)

where s_{α} = 0 if α ∉ ℤ and s_{α} = 1 if α ∈ ℤ. The spectrum of f is defined by

Sp(f) = ∑ (α) ∈ ℤ[ℚ]

where the sum is over the spectral pairs (α, ω). In the local case the spectrum has three important properties: symmetry, Sebastiani–Thom property, and semicontinuity. In the sequel, we will discuss these properties briefly in our global situation (i.e. for (∗)-polynomials).

The symmetry of the weight filtration gives h_{λ}^{pq} = h_{λ}^{n+s−q,n+s−p}, hence the invariance of Spp(f) with respect to the transformation (α, n + k) ↔ (α + k, n − k). The complex conjugation (i.e. h_{λ}^{pq} = hqp ; λ̄) composed with the first symmetry gives the second one: (α, n + k) ↔ (n − 1 − α, n − k). This shows also that the spectrum is symmetric with respect to (n − 1)/2 (for details in the local situation, see e.g. [1, 13.3.C]). Moreover, we have also a (global) Sebastiani-Thom theorem: if (α, ω), (α′, ω′) ∈ ℚ × ℕ, define (α, ω) ∗ (α′, ω′) = (α + α′ + 1, ω + ω′ + 1), and extend ∗ to ℤ[ℚ × ℕ] by linearity. Set:

S_{d} = ∑_{0<s<d} (−s/d, 0) ∈ ℤ[ℚ × ℕ].

(6.1) Theorem: If f is a (∗)-polynomial of degree d, then

Spp(f + x^{d}) = Spp(f) ∗ S_{d}.

Proof: For the spectral pairs of f + x^{d} corresponding to eigenvalue 1 the result is easy, it uses (3.1.a) and the fact that

P^{n}(X′_{0}) = ⊕_{ξ^{d}=1, ξ≠1} P^{n}(X′_{0})_{ξ}.

For eigenvalues ξ ≠ 1 with ξ^{d} = 1 the result follows from (3.1.b) and (4.10) (cf. the proof of (5.7) ). For ξ^{d} ≠ 1 one uses (3.1.b) and the local Sebastiani-Thom theorem ([18]). In this later case the computation is rather long and tedious. The details are left to the reader. (For a different proof, in the context of “cohomologically tame” polynomials, see [14]).

The main application of (6.1) is a semicontinuity property of Spp(f). If f is a (∗)-polynomial, we will consider deformations f_{λ} : ℂ^{n+1} → ℂ of f with λ ∈ (ℂ, 0) and fixed degree d. Notice that since the (∗) condition defines an open dense subset in the space of polynomials in n + 1 variables of degree d, f_{λ} is also a (∗)-polynomial (Actually, because of (3.2.d) it would be enough to consider deformations f_{λ} = (f_{d})_{λ} +f_{d−1} +…, where (f_{d})_{λ} is a deformation of f_{d}). Since in such a deformation one has μ^{∞}(f_{λ≠0}) ≥ μ^{∞}(f) (where μ^{∞}(f) denotes the dimension of the middle cohomology group of the generic fibre of f), one cannot expect an upper semicontinuity for the spectrum as in the local case, but a lower semicontinuity.

(6.2) Definition (cf. [25]): Given a deformation (f)_{λ} of a (∗)-polynomial f and a subset A ⊆ ℝ, we will say that A is a lower semicontinuity domain for (f)_{λ} if the function which associates to λ ∈ (ℂ, 0) the sum of the frequencies of spectral numbers of f_{λ} in A (denoted by s_{A}(f_{λ}) ) is lower semicontinuous, i.e.

s_{A}(f_{λ≠0}) ≥ s_{A}(f).

(6.3) Theorem: Let f be a (∗)-polynomial of degree d. Every half open interval (\frac{k}{d}, \frac{k}{d} +1], k ∈ ℤ, is a lower semicontinuity domain for deformations of f with fixed degree.

Proof: By (E.0) and (4.5), there is a C ∈ ℕ depending only on n, d, p such that dim Gr_{F}^{p} H^{n}(X_{t}^{0}) = C − dim Gr_{F}^{p} ⊕_{j} H^{n−1}(F_{j}). By the local uppersemicontinuity result ([25]) the theorem follows for the interval (t, t + 1], t ∈ ℤ. This, together with (6.1), gives the result similarly as in the local case (see [loc.cit, (2.7)]). (6.4) Remarks: a) Actually, by a tedious combinatorial argument, using the corresponding local result and (3.1), one can show that any interval (t, t + 1], t ∈ ℝ is a lower semicontinuity domain. We do not emphasize this proof because a more conceptual proof is in preparation, which works for any cohomologically tame polynomial ([14]).

b) The symmetry shows that [t, t + 1) is also a semicontinuity domain.

c) As in the local case, (5.12) has the following corollary: Let Spp (f) ∈ ℤ[ℚ × ℕ] be the set of spectral pairs of a (∗)-polynomial. Let Spp_{mod−2}(f) be its projection in ℤ [ℚ/2 ℤ×ℕ]. The knowledge of Spp_{mod−2}(f) is equivalent to that of the real Seifert form of the Milnor fibration of f at infinity (cf. [10]).

7. Examples.

(7.1) Case n = 1.

Let f ∈ ℂ[X, Y] be a (∗)-polynomial with highest degree form f_{d}. Write

f_{d} = ∏_{j=1}^{m} l_{j}^{α_{j}}

where the l_{j} ∈ ℂ[X, Y] are distinct linear forms and ∑ α_{j} = d. Set α = gcd(α_{1}, …, α_{m}). For x ∈ ℝ, set δ(x) = 1 if x ∈ ℤ, δ(x) = 0 if x ∉ ℤ. Denote by ⌈x⌉ the smallest integer bigger or equal than x.

The equivariant Hodge numbers of P^{2}(X′_{0}) can be computed directly, if ξ = e(s/d) with 0 < s < d then h^{1,1}(P^{2}(X′_{0})_{ξ}) = δ(\frac{sα}{d}), the other numbers are zero. Then from theorem (4.7) (using also [1, p. 305, Théorème 6] or [21] for the local terms appearing in the formulas in (4.7)) one gets all equivariant Hodge numbers of P^{1}(X′_{0}). Now one computes the equivariant Hodge numbers of the MHS at infinity using (3.1). One gets:

a) p_{1}^{1,1}(f) = m − 1. b) If ξ = e(−s/d), 0 < s < d, then

p_{ξ}^{11}(f) = −δ(\frac{sα}{d}) + ∑_{j} δ(\frac{sα_{j}}{d}).

p_{ξ}^{0,1}(f) = −s − 1 + δ(\frac{αs}{d}) + ∑_{j}⌈\frac{sα_{j}}{d} ⌉.

p_{ξ}^{1,0}(f) = s − 1 + δ(\frac{αs}{d}) − ∑_{j}[\frac{sα_{j}}{d}].

c) If ξ^{d} ≠ 1, put ξ = e(−β) with 0 < β < 1 and set γ = {(d − 1)β}. Notice that β + γ ∉ ℤ. If ξ^{d−1} = 1, then p_{ξ}^{∗,∗}(f) = 0. If ξ^{d−1} ≠ 1 and β + γ < 1, then p_{ξ}^{10}(f) = 0 and p_{ξ}^{0,1}(f) = #{j | ξ^{(d−1)α_{j}} = 1}. If ξ^{d−1} ≠ 1 and β + γ > 1, then p_{ξ}^{01}(f) = 0 and p_{ξ}^{1,0}(f) = #{j | ξ^{(d−1)α_{j}} = 1}. (All other primitive equivariant Hodge numbers vanish.)

Example: If f = x^{2}y^{2} + (x + y)^{3}, then p_{1}^{11} = 1, p_{−1}^{11} = 1 hence h_{−1}^{00} = 1 too, p_{e(−1/6)}^{01} = p_{e(−5/6)}^{10} = 2. In particular, the rank of the midle (co)homology of the generic fiber is 7, and the monodromy at infinity has a Jordan block of size two.

(7.2) Zariski’s examples.

Let f_{6} ∈ ℂ[X, Y, Z] be a homogeneous polynomial of degree 6 which defines a sextic in ℙ^{2} with six cusps and no other singularities. Let f = f_{6} + … be any (∗)- polynomial with highest degree form f_{6}. The Hodge numbers of X^{∞} = {f_{6} = 0} ⊂ ℙ^{2} are easy to compute. Those of X′_{0}, the 6-fold cyclic covering of ℙ^{2} branched along X^{∞}, depend on the position of the cusps. The equivariant Hodge numbers of P^{3}(X′_{0}) can be computed with the aid of [3, ch.6,(4.9), see also (3.18.ii)] One has two cases:

Case i) If the six cusps are on a conic then

h_{e(1/6)}^{2,1}(P^{3}(X′_{0})) = 1 = h_{e(5/6)}^{1,2}(P^{3}(X′_{0})).

Case ii) If the six cusps are not on a conic then P^{3}(X′_{0}) = 0.

Now from theorem (4.7) (and using again the results in [21] or [1, p. 305, Théorème 6] to compute the local terms appearing in (4.7)), one gets the equivariant Hodge numbers of P^{2}(X′_{0}). Finally, theorem (3.1) gives the primitive equivariant Hodge numbers of the MHS at infinity of f. One has:

a) Both in cases (i) and (ii),

p_{1}^{1,2}(f) = p_{1}^{2,1}(f) = 4.

b) If ξ = e(s/6) with 0 < s < 6 the numbers p^{p,q}(f)_{ξ} are:

(p, q) (1, 1) (2, 0) (0, 2) (1, 2) (2, 1)
s = 5 4 (i) 0 1 (i) 5 (i) 0
3 (ii)  0 (ii) 6 (ii)
s = 4  6  0  3  0  0
s = 3  7  1  1  0  0
s = 2  6  3  0  0  0
s = 1 4 (i) 1 (i)  0  0  5 (i)
3 (ii) 0 (ii)  6 (ii)

c) Both in cases (i) and (ii), if ξ = e(l/30) with l ∈ {1, 7, 11, 13, 17, 19, 23, 29} then p_{ξ}^{1,1}(f) = 6.

All other primitive equivariant Hodge numbers vanish.

References

[1] V. Arnold, A. Varchenko and S. Goussein-Zadé. Singularités des Applications Différentiable, 2^{e} partie. Éditions Mir Moscou, 1986.

[2] E. Cattani and A. Kaplan. Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure. Invent. Math., 67:101–115, 1982.

[3] A. Dimca. Singularities and Topology of Hypersurfaces. Universitext. Springer Verlag, 1992.

[4] A. Dimca. Hodge Numbers of Hypersurfaces. Abh. Math. Sem. Univ. Hamburg, 66:377–386, 1996.

[5] R. García López and A. Némethi. On the monodromy at infinity of a polynomial map. Compos. math., 100:205–231, 1996. Appendix by R. García López and J. Steenbrink.

[6] R. García López and A. Némethi. On the monodromy at infinity of a polynomial map, II. Compos. math., 115:1–20, 1999.

[7] P. Griffiths On the periods of certain rational integrals, I,II. Annals of Math., 90:460– 541, 1969.

[8] H. A. Hamm. Hodge numbers for isolated singularities of nondegenerate complete intersections. In Singularities (Oberwolfach, 1996). Progress in Math., 162, pp. 37– 60. Birkhäuser, Basel, 1998.

[9] V. Navarro Aznar. Sur la théorie de Hodge-Deligne. Invent. Math., 90:11–76, 1987.

[10] A. Némethi. The real Seifert form and the spectral pairs of isolated hypersurface singularities. Compos. Math., 98:23–41, 1995.

[11] A. Némethi. On the Seifert form at infinity associated with polynomial maps. J. Math. Soc. Japan, 51:63–70, 1999.

[12] A. Némethi. The semi-ring structure and the spectral pairs of sesqui-linear forms. Algebra Colloq., 1:85–95, 1994.

[13] A. Némethi. The mixed Hodge structure of a complete intersection with isolated singularity. C.R. Acad. Sci. Paris, t.321, Série I, pp. 447–452, 1995.

[14] A. Némethi and C. Sabbah. Semicontinuity of the spectrum at infinity. pre-print.

[15] F. Pham. Vanishing homologies and the n variable saddlepoint method. In Proc. Symp. Pure Math., vol. 40, 319–333, 1983.

[16] C. Sabbah. Hypergeometric periods for a tame polynomial. pre-print.

[17] M. Saito. Mixed Hodge modules. Publ. RIMS Kyoto Univ. 26, 221-333, 1990.

[18] J. Scherk and J.H.M. Steenbrink. On the Mixed Hodge Structure on the Cohomology of the Milnor Fibre. Math. Ann., 271:641–665, 1985.

[19] W. Schmid. Variation of Hodge structures: the singularities of the period mapping. Invent. Math. 22, 211-319, 1973.

[20] J.H.M. Steenbrink. Limits of Hodge Structures. Inv. Math., 31:229–257, 1976.

[21] J.H.M. Steenbrink. Intersection form for quasi-homogeneous singularities. Compos. Math., 34:211–223, 1977.

[22] J.H.M. Steenbrink. Mixed Hodge structure on the vanishing cohomology. In Real and Complex Singularities, Oslo 1977, pages 397–403, Alphen a/d Rhijn, 1977. Sijthoff & Noordhoff.

[23] J.H.M. Steenbrink. Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math., vol. 40, pages 513–536, 1983.

[24] J.H.M. Steenbrink and S. Zucker. Variation of mixed Hodge structure. I. Invent. Math., 80:489–542, 1985.

[25] J.H.M. Steenbrink. Semicontinuity of the singularity spectrum. Invent. Math., 79:557–565, 1985.

Universidad de Barcelona, Dept. de Algebra y Geometría. Gran Via, 585, 08007 Barcelona, Spain. e-mail address: rgarcia@cerber.mat.ub.es The Ohio State University, 231 West 18th Avenue, Columbus OH 43210, USA. e-mail address: nemethi@math.ohio-state.edu