On the connectivity of complex affine hypersurfaces, II

by Alexandru Dimca and Laurentiu Paunescu

Let f ∈ ℂ[x_{0}, x_{1}, ..., x_{n}] be a polynomial and w = (w_{0}, ..., w_{n}) a system of positive integer weights. Consider f = f_{0} +...+f_{e} +f_{d} the decomposition of f as a sum of weighted homogeneous components f_{i} where deg(f_{i}) = i with respect to w. Here f_{d} is the top degree nonzero component and we assume that f_{j} = 0 for e < j < d for some integer e > 0.

For any polynomial g we denote by ∂g its gradient

∂g = (g_{x_{0}}, ..., g_{x_{n}})

where g_{x_{i}} is the partial derivative of g with respect to x_{i}.

Define the subset S(f) in ℂ^{n+1} given by the equations ∂f_{d} = 0, f_{e} = 0. With this notation our main result is the following.

THEOREM

(i) Let Σ(f) = {x ∈ ℂ^{n+1}; ∂f = 0} be the set of singular points of the polynomial f. Then dimΣ(f) ≤ dimS(f).

(ii) Any fiber of the polynomial f is (n − 1−dimS(f))-connected.

Improving previous results on the connectivity of affine hypersurfaces, the first author has proved the part (ii) of this result in the case of the usual weights w = (1, 1, ..., 1) in [D1].

The proof there was based on the properties of tame and quasi-tame polynomials introduced by Broughton [B], Némethi [N1], [N2] and Némethi-Zaharia [NZ]. More precisely, the general case was reduced to these special classes of polynomials by applying an affine Lefschetz hyperplane section theorem due to Hamm [H]. This proof cannot be applied for arbitrary weights since in general we don’t have enough weighted hyperplanes to proceed by induction (even though Lemma 9. in [D1] can be easily extended to show that the polynomial f is quasitame when dimS(f) = 0).

On the other hand, Némethi and Sabbah [NS] have recently studied tame polynomials defined on affine varieties, but their definition does not seem easily adaptable to keep track of the chosen weights.

Our proof below is based on a powerful general result by Hamm, namely Proposition 3 in [H] (which is the key step in his proof of the affine Lefschetz theorem on hyperplane sections and which we slightly correct in Lemma 3. below) and on basic properties of isolated complete intersection singularities, for which we refer the reader to Looijenga’s book [L].

The following example shows the strength of the new result over the old.

Example 1.

Let f : ℂ^{6} → ℂ be the polynomial given by

f(x) = x_{0}^{3}x_{1}^{3}x_{2}^{3} + (x_{0} + x_{1} + x_{2})^{7} + x_{3}^{5} + x_{4}^{4} + x_{5}^{3} + (x_{0} + x_{1} + x_{2} + x_{3} + x_{4} + x_{5})^{2}

If we consider the usual weights w = (1, 1, ..., 1), then dimS(f) = 4, and hence we get that the fibers of f are 0-connected, i.e. they are connected. In this case d = 9, e = 7

Now, if we use the system of weights w = (20, 20, 20, 36, 45, 60) chosen such that the top degree form f_{d} contains as many monomials as possible, then dimS(f) = 1 hence the fibers of f are in fact 3-connected. In this case d = 180, e = 140.

It is interesting to notice that the topological result above has some useful algebraic consequences expressed in terms of various complexes of differential forms associated naturally to the polynomial f.

To state them, let A^{∗} = (Ω^{∗},d) denote the De Rham complex of global regular differential forms on ℂ^{n+1} with d the exterior differentiation acting on forms (not to be confused to the degree d which occured above).

The first complex associated to f is the complex K_{f}^{∗} = (Ω^{∗},df∧) which can be identified to the Koszul complex of the partial derivatives of f in the polynomial ring ℂ[x_{0}, ...x_{n}] = Ω^{0}.

The De Rham complex A^{∗} has a natural subcomplex B_{f}^{∗} = (df ∧ Ω^{∗},d) and a natural quotient complex C_{f}^{∗} = A^{∗}/B_{f}^{∗} called the complex of global relative differential forms.

Finally, one can consider as in Dimca-Saito [DS], the mixture of De Rham and Koszul complexes, namely the complex D_{f}^{∗} = (Ω^{∗}, d−df∧).

COROLLARY.

Let f be a polynomial as above and assume that k =dimS(f) < n. Then

(i) H^{i}(K_{f}^{∗}) = 0 for all 0 ≤ i ≤ n − k; (ii) H^{1}(B_{f}^{∗}) = ℂ[f]df and H^{i}(B_{f}^{∗}) = 0 for all i ≠ 1, 0 ≤ i ≤ n − k; (iii) H^{0}(C_{f}^{∗}) = ℂ[f] and H^{i}(C_{f}^{∗}) = 0 for all 0 < i ≤ n − k − 1; (iv) H^{i}(D_{f}^{∗}) = 0 for all 0 ≤ i ≤ n − k.

Proof of the Theorem

Let N be any common multiple of the weights w_{i} and consider the integers m_{i} = N/w_{i}. Let ϕ : ℂ^{n+1} → ℂ^{2n+2} be the embedding

ϕ(x) = (x_{0}, ..., x_{n}, x_{0}^{m_{0}}, ..., x_{n}^{m_{n}})

Let X = f^{−1}(0). It is enough to prove that

(i′) dimX_{sing} ≤ dimS(f); (ii′) X is (n − 1−dimS(f)) connected.

Let Y = ϕ(X) and note that ϕ : X → Y is an isomorphism. Let k =dimS(f) and let H_{1} : l_{1} = 0,...,H_{k} : l_{k} = 0 be generic hyperplanes in ℂ^{2n+2}.

Using repeatedly Lefschetz hyperplane section theorem in [H] we get that the inclusion Y_{0} = Y ∩ H_{1}... ∩ H_{k} → Y is an (n − k)-equivalence.

Let X_{0} = ϕ^{−1}(Y_{0}) and g_{i}(x) = l_{i}(ϕ(x)) be the corresponding pull-back polynomials on ℂ^{n+1}. Note that the weighted homogeneous component g_{i,N} is a polynomial of Pham-Brieskorn type, i.e. a sum ∑ a_{ij}x_{j}^{m_{j}}.

Lemma 1.

(a) A = {x ∈ ℂ^{n+1}; g_{1}(x) = ... = g_{k}(x) = 0} is a smooth complete intersection of dimension n + 1 − k;

(b) the set {x ∈ ℂ^{n+1}; x ∈ S(f), g_{1,N}(x) = ... = g_{k,N}(x) = 0} is just the origin;

(c) there exist weighted homogeneous polynomials h_{1}, ..., h_{n} such that the germs at the origin of the sets {x ∈ ℂ^{n+1}; g_{1,N}(x) = ... = g_{k,N}(x) = f_{d}(x) = h_{1}(x) = ... = h_{j}(x) = 0} is an isolated complete intersection singularity of dimension max(n − k − j, 0) for all j = 1, ..., n.

(d) (Y_{0})_{sing} = Y_{sing} ∩ H_{1} ∩ ... ∩ H_{k} and dim(Y_{0})_{sing}=dimY_{sing} − k.

Proof of Lemma 1.

All these claims follow by general transversality arguments involving the parameter space of all the coefficients a_{ij}.

More precisely, to have (a) we take the hyperplanes H_{i} such that their intersection is transverse to the smooth variety ϕ(ℂ^{n+1}).

For (b), consider the composition ψ = p_{2} ∘ ϕ, where p_{2} : ℂ^{2n+2} → ℂ^{n+1}, (x, y) ↦ y is the projection on the last (n + 1) coordinates.

Then B = ψ(S(f)) is an algebraic set in ℂ^{n+1} of dimension at most k. It follows that the intersection of k generic hyperplanes H′_{i}, i = 1, ..., k, cuts B only at the origin. We can use the coefficients of the hyperplane H′_{i} to construct our hyperplane H_{i}, i.e. the corresponding coefficients of the y-variables coincide.

The last claims (c) and (d) follow by similar easy arguments.

Note that (i′) in our Theorem follows from Lemma 1. (d) in view of the next result.

Lemma 2.

For any a ∈ ℂ the variety X_{a} = {x ∈ A; f(x) = a} has at most isolated singularities.

Proof of Lemma 2.

If X_{a} has non-isolated singularities, then an application of the curve selection lemma, see Milnor [M], would produce a path p : (0, ϵ) → (X_{a})_{sing} given by a Laurent power series

p(t) = c_{0}t^{s} + c_{1}t^{s+1} + ...

where c_{i} ∈ ℂ^{n+1}, c_{0} ≠ 0 and the integer s is strictly negative.

Note that we can reparametrize this path by replacing t with t^{N}, i. e. we can suppose that all exponents in p(t) with non zero coefficients are divisible by our weights w_{i}.

There is a standard ℂ^{∗}-action on ℂ^{n+1} associated with these weights, namely

u ∗ x = (u^{w_{0}}x_{0}, ..., u^{w_{n}}x_{n})

We can rewrite our path using this action in the form

p(t) = t^{s} ∗ c_{0} + t^{s+1} ∗ c_{1} + ...

where the coefficients c_{i}’s and the exponent s are different in general from those considered first above, but which enjoy the same properties, i.e. c_{0} ≠ 0 and s < 0.

Next we have g_{i}(p(t)) = g_{i,N}(c_{0})t^{sN} + ... = 0 where the dots represent higher order terms, hence we have g_{i,N}(c_{0}) = 0. In a similar way we get ∂f_{d}(c_{0}) = 0.

Due to the Euler relations kf_{k} = ∑_{i=0,n} w_{i}f_{k,x_{i}} we have also

0 = df(p(t)) − ∑_{i=0,n} w_{i}f_{x_{i}}(p(t))p_{i}(t) = (d − e)f_{e}(p(t)) + ... = (d − e)f_{e}(c_{0})t^{se} + ... which implies that f_{e}(c_{0}) = 0, a contradiction with the property (b) above.

This finishes the proof of the Lemma and of part (i) in our Theorem.

To continue the proof, we need the following result.

Lemma 3.

Let A be a smooth complete intersection in ℂ^{p} with dimA = m. Let f_{1}, ..., f_{p} be polynomials in ℂ[x_{1}, ..., x_{p}] such that the following conditions hold.

(c0) The set {z ∈ A; |f_{1}(z)| ≤ a_{1}, ..., |f_{p}(z)| ≤ a_{N}} is compact for any positive numbers a_{j}, j = 1, ..., p.

(c1) The connected components of the critical set of the function f_{1} : A → ℂ are compact.

(cj) (for j = 2, ..., p) Let F be the critical set of the mapping (f_{1}, ..., f_{j}) : A → ℂ^{j} for some j. Then (f_{1}, ..., f_{j−1}) : F → ℂ^{j−1} is a proper mapping.

Then A has the homotopy type of a space obtained from A_{1} = {x ∈ A; f_{1}(x) = 0} by adjoining m-cells.

Proof of Lemma 3.

Apart from the condition (c0) this lemma is a special case of Proposition 3. in Hamm [H]. The following example shows that in fact the condition (c0) is necessary. An inspection of the proof in [H] shows that with this extra condition his Proposition 3. and hence our Lemma 3. are true. More precisely, the condition (c0) allows one to use the Morse theory for the function |f_{1}|^{2} to increase a_{1} for some fixed (and very large) a_{2}, ..., a_{p} as in Hamm’s proof.

Example 2.

Let A be ℂ^{2}, f_{1} = x^{2}y + x and f_{2} = x. Then all the assumptions in Proposition 3. in [H] are fulfilled, but not the conclusion, i.e. ℂ^{2} cannot be obtained from the non connected space f_{1}^{−1}(0) by adjoining 2-cells.

We return to the proof of our Theorem. To prove the claim (ii′) we use Lemma 3. for the following data:

The set A is the smooth complete intersection A = {x ∈ ℂ^{n+1}; g_{1}(x) = ... = g_{k}(x) = 0}, see property (a) above.

The polynomials f_{i} which appear in Lemma 3. are chosen as follows f_{1} = f and f_{j} = h_{j−1} for j = 2, ..., n + 1 = p, with the polynomials h_{j} from Lemma 1. (c).

These polynomials satisfy the following conditions:

(c0) as follows easily from Lemma 1. (c), (c1) as follows from Lemma 2. and (cj) as we explain now. In our case, a critical point for (f_{1}, ..., f_{j}) : A → ℂ^{j} is just a

singular point of the variety Z given by the equations

g_{1}(x) = ... = g_{k}(x) = 0, f(x) = a_{1}, h_{1}(x) = a_{2}, ..., h_{j−1}(x) = a_{j}

This variety corresponds to a fiber in a deformation of the isolated complete intersection singularity

g_{1,N}(x) = ... = g_{k,N}(x) = f_{d}(x) = h_{1}(x) = ... = h_{j−1}(x) = 0

Indeed, we have the following general localization result, see also [D2], p. 157 and p. 161 (the case discussed there is for the usual weights w = (1, ..., 1)).

Lemma 4.

Let Z be any affine variety in ℂ^{n+1} given as the zero set of the polynomials P_{j}(x) = Q_{j}(x) + R_{j}(x) for j = 1, ..., m where degP_{j}(x) =degR_{j}(x) = d_{j}, degQ_{j}(x) < d_{j} and the polynomials R_{j} are weighted homogeneous with respect to the weights w (the degrees are relative to these weights as well).

Then Z is homeomorphic to a fiber in a deformation of the weighted homogeneous singularity (Z_{0}, 0) : R_{1}(x) = ... = R_{m}(0) = 0.

Proof of Lemma 4. For any real number r > 0 we set y = x/r and define

Z_{r} = {y ∈ ℂ^{n+1}; P_{j}(r ∗ y) = 0 for j = 1, ..., m}

Multiplication by r induces a homeomorphism x ↦ r ∗ x between Z_{r} ∩ B_{e} and Z ∩ B_{er} where B_{a} = {x ∈ ℂ^{N}; |x_{0}|^{2m/w_{0}} + ... + |x_{n}|^{2m/w_{n}} < a} is a weighted open ball for any a > 0.

Note that Z_{r} ∩ B_{e} is the zero set of the polynomials P_{j,r}(y) = R_{j}(y) + Q_{j}(r ∗ y)r^{−d_{j}}. For r >> 1/e large enough and e small, the following two claims hold.

(a1) P_{j,r} is a small deformation of the polynomial P_{j}, i.e. Z_{r} ∩ B_{e} is a fiber in a deformation of the singularity (Z_{0}, 0).

(a2) Z is homeomorphic to Z ∩ B_{er}, see [D2], p. 26 for a similar result.

This finishes the proof of Lemma 4.

Back to checking the conditions (cj), we see by Lemma 4. (or rather by its proof) that any such variety Z is either smooth or has isolated singularities. Projecting on the space ℂ^{j−1} means that now we consider a family of fibers as above corresponding to a line in the base of the deformation of our singularity. If h_{j} is chosen general enough, this line is not contained in the discriminant Δ of this deformation, i.e. there is a finite number of singular fibers when we vary a_{j}. This follows from the basic fact that a nonconstant regular function on a variety with isolated singularities has finitely many critical values.

It follows that the map (f_{1}, ..., f_{j−1}) : F → ℂ^{j−1} is the composition of two finite maps, the first one from F to the discriminant Δ of the deformation and the second one the projection Δ → ℂ^{j−1}. This explains why the condition (cj) holds as well in our setting for j > 1.

By Lemma 3. we have that the inclusion X_{0} → A in an (n − k)-equivalence. As above, we can identify A with a smooth fiber (Milnor fiber) in the deformation of the corresponding isolated complete intersection, hence A is (n−k)-connected being a bouquet of (n − k + 1)-spheres.

Combining this with the (n − k)-equivalence X_{0} → X obtained at the beginning, we have that X is (n − k − 1)-connected.

This ends the proof of our Theorem.

Proof of the Corollary

The first claim (i) depends only on part (i) of our Theorem. Indeed, the cohomology groups of the Koszul complex are finitely generated ℂ[x_{0}, ..., x_{n}]-modules and to prove that one of them is trivial it is enough to show that all its localizations at maximal ideals are trivial.

To check this local property we can use GAGA and replace algebraic localization by analytic localization. At this level the result follows from Greuel’s generalized version of the de Rham Lemma, see [G], (1.7) or from a general result in Looijenga’s book [L], namely Corollary (8.16), p. 157 (take X a smooth germ and k = 1 in that statement).

To prove (ii) and (iii), we consider the exact sequence of complexes

0 → B_{f}^{∗} → A^{∗} → C_{f}^{∗} → 0

This shows that it is enough to prove (ii). We have in fact the following more precise result. By convention, the dimension of the empty set is taken to be −1.

Lemma 5.

Let f : ℂ^{n+1} → ℂ be a non-constant polynomial function. Then the following statements are equivalent for any positive integer k:

(i) H^{1}(B_{f}) = ℂ[f]df and H^{i}(B_{f}) = 0 for i ≠ 1, i ≤ n − k;

(ii) the reduced cohomology groups H̃^{i}(F_{t}, ℂ) are trivial for 0 ≤ i ≤ n − k − 1 for F_{t} the general fiber of the polynomial f.

Proof of Lemma 5.

To simplify notation and make it similar to the notation in [DS1] for easy reference, in this proof we set X = ℂ^{n+1} and S = ℂ. The algebraic Gauss-Manin system of f : X → S is the direct image G_{f} = f_{+}O_{X}[−n − 1] of the D_{X}-module O_{X}, see Borel [Bo]. We shift it by (−n − 1) to get a complex in positive degrees, as it is more usual in algebraic topology.

At the level of global sections on S, the algebraic Gauss-Manin system of f is represented by the complex of A_{1} = ℂ[t] < ∂ >-modules G_{f}^{∗} = (Ω^{∗}[∂], d_{f}) where the ℂ -linear differential d_{f} is defined by d_{f}(ω∂^{m}) =dω∂^{m}−df ∧ ω∂^{m+1}, see [S], [DS1] and [DS2] for more on this complex.

The cohomology sheaves G_{f}^{i} = H^{i}(G_{f}) are regular holonomic D_{S}-modules and the Riemann-Hilbert correspondence [Bo] implies that DR_{S}(G_{f}^{i}) =^{p} R^{i}f_{∗}ℂ_{X}.

Here ^{p}R^{i}f_{∗}ℂ_{X} =^{p} H^{i}(f_{∗}ℂ_{X}) and ^{p}H^{i} denotes the perverse cohomology functor, see [BBD]. Basic properties of perverse sheaves gives the following isomorphism

(∗)  ^{p}R^{i}f_{∗}ℂ_{X} = R^{i−1}f_{∗}ℂ_{X}[1]

see [S] or [DS2] for a proof.

Let R^{i} = R^{i}f_{∗}ℂ_{X}. Then the fiber R_{y}^{i} is isomorphic to the cohomology group (with ℂ coefficients) H^{i}(T (F_{y})) where T (F_{y}) is the tube about the fiber F_{y} = f^{−1}(y) defined by T (F_{y}) = f^{−1}(D_{y}), with D_{y} a small open disc centered at y.

Let Λ be the finite bifurcation set for f, i.e. f : X \ f^{−1}(Λ) → S \ Λ is a locally trivial smooth fibration.

For y ∉ Λ we then have R_{y}^{i} = H^{i}(F_{y}) (F_{y} is the general fiber of f by definition) so the condition (ii) implies the vanishing of these stalks for 1 ≤ i ≤ n − k − 1. To obtain the vanising of the whole sheaf R^{i} we need the following vanishing result.

H_{Λ}^{0}(S, R^{i}) = 0

for all i > 0, i.e. there are no sections in these sheaves with supports only at the finite set Λ. To see this, note that the vector space H_{Λ}^{0}(S, R^{i}) is a subspace of H^{0}(S, R^{i}) which vanishes due to the E_{2}-term in the Leray Spectral sequence of f : X → S. The only nontrivial fact here is that E_{2}^{p,q} = H^{p}(S, R^{q}) = 0 for p ∉ {0, 1} by Artin vanishing Theorem, see [SGA4], XIV, 3.2 and XVI, 4.1.

In this way (ii) implies the following condition

(ii′)  R^{0}f_{∗}ℂ_{X} = ℂ_{S} and R^{i}f_{∗}ℂ_{X} = 0

for 0 < i < n − k.

An alternative, more geometric way to prove the implication (ii) ⟹ (ii′) is to use the same idea as in Lemma 1. in [ACD].

The converse implication is obvious, so (ii′) is equivalent to (ii).

Since the de Rham functor is faithful (i.e. DR_{S}(M) = 0 iff M = 0) it follows easily that the condition (ii′) above is equivalent to

(ii′′) H^{1}(G_{f}^{∗}) = ℂ[f]df and H^{i}(G_{f}^{∗}) = 0 for i ≠ 1, i ≤ n − k.

The complex G_{f}^{∗} comes equipped with a decreasing filtration given by

F^{s}G_{f}^{m} = Ω^{m}[∂]_{≤m−s}

where the filtration on the right hand side is by the degree with respect to ∂ (Sabbah prefers to work with a similar but increasing filtration in [S]).

The general theory of spectral sequences, see if necessary [Mc], associates to this decreasing, exhaustive and bounded below filtration a spectral sequence with E_{1}^{s,t} = H^{s+t}(Gr_{F}^{s} G_{f}^{∗}) converging to H^{s+t}(G_{f}^{∗}).

For t > 0, we have E_{1}^{s,t} = H^{s+t}(K_{f}^{∗}) and hence in particular in our case we have E_{1}^{s,t} = 0 for all t > 0, s + t ≤ n − k by our Corollary (i) above.

Moreover, the terms E_{1}^{s,0} with the corresponding differential d_{1} : E_{1}^{s,0} → E_{1}^{s+1,0} coming from the spectral sequence can be identified for s < n − k (since we need again Corollary (i)) to the corresponding initial part in the complex B_{f}^{∗}.

Since this part of the spectral sequence clearly degenerates at the E_{2}-term, i.e. E_{2}^{s,0} = E_{∞}^{s,0} for s ≤ n − k, we obtain the equivalence (i) ⇔ (ii′′) which completes the proof of Lemma 2.

To end the proof of our Corollary, we have just to use the main results in [DS1], saying that the cohomology of the complex D_{f}^{∗} twisted by −1 is just the reduced cohomology of the general fiber of the polynomial f.

Note that we have shown in fact the equivalence of the conditions (ii), (iii) and (iv) in our Corollary for any polynomial f.

Remarks.

(I) The condition k = dimΣ(f) < n is equivalent to saying that the fibers of f are all reduced. This condition is not needed for part (i) of our Corollary (which holds even when k = n), but it is necessary for parts (ii)-(iv) since it implies that the general fiber of f is connected.

Indeed, it is well known that any polynomial f can be written as a composition f(x) = h(g(x)) where g : ℂ^{n+1} → ℂ has a connected general fiber and h : ℂ → ℂ, both g and h being polynomials. Since it is difficult to locate a reference for this fact, here is a short proof of it suggested to the first author by Zaidenberg, to whom we are grateful.

Assume that f is not a constant polynomial and let f̃: X → ℙ^{1} be a smooth compactification of f. Then the Stein Factorization Theorem, see [Har], p.280, gives a smooth curve C and morphisms g̃ : X → C and h̃ : C → ℙ^{1} such that h̃ ∘ g̃ = f̃ and such that all the fibers of g̃ are connected.

A generic line L in ℂ^{n+1} has the following properties:

(i) f is not constant when restricted to L;

(ii) the closure L̃ of L in X is a smooth rational curve which meets f̃^{−1}(∞) at exactly one point and this intersection is transverse. Then L̃ is a rational curve and g̃|L̃ is a non constant map. This implies that C = ℙ^{1}. Moreover, we have h̃^{−1}(∞) = ∞, otherwise the condition (ii) above is contradicted. This implies that h̃ gives by restriction to ℂ = ℙ^{1} \ {∞} a morphism h : ℂ → ℂ such that f = h ∘ g where g = g̃|ℂ^{n+1} has its general fiber connected.

(II) For a given polynomial f : ℂ^{n+1} → ℂ and a given system of weights w we define the connectivity order of f with respect to w to be the integer

c_{w}(f) = n − 1 − k

where k =dimS(f).

Then it is easy to see that for the sum f(x)+g(y) of two polynomials f ∈ ℂ[x_{0}, ..., x_{n}] and g ∈ ℂ[y_{0}, ..., y_{m}] on two independent groups of variables and for any weights w′ associated to g such that the corresponding degrees d for f and g coincide (by multiplying any weights w for f and w′ for g by suitable positive integers this can be always achieved) we have

c_{w,w′}(f + g) ≥ c_{w}(f) + c_{w′}(g) + 1

in spite of the fact that in general the corresponding degrees e will be different.

Note that the general fiber of f + g is the join of the general fibers of f and g, see [N3], but nothing is known about the special fibers.

(III) Siersma and Tibăr have shown in [ST] that in the case of the usual weights w = (1, 1, ..., 1) and if e = d − 1 and dimΣ(f) <dimS(f) = k, then the general fiber of f is (n − k)-connected, i.e. a better by 1 estimate than that given by our Theorem. They also give an example showing that this better estimate fails for the special fibers.

Improving the results in [ST], Tibăr has recently obtained very general connectivity results for the fibers of polynomial mappings, see [T], Thm . (5.5). However the effectiveness of his results depends on the explicit construction of fiber-compactifying extensions of the polynomial function f having a small critical set at infinity. It seems to us that such constructions are difficult to handle for general weights w or even for the usual weights when e < d − 1.

Acknowledgements.

The authors thank the referee for his very useful remarks on the presentation and style of this paper.

The first author would like to thank Dan Burghelea and András Némethi for their kind invitation to visit Ohio State University at Columbus.

The second author thanks the staff of the Tokyo Institute of Technology for their hospitality.

This research was supported in part by the ARC Grant A69530267.

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Laboratoire de Mathématiques Pures de Bordeaux Université Bordeaux I 33405 Talence Cedex, FRANCE

email: dimca@math.u-bordeaux.fr

School of Mathematics and Statistics, Sydney University, Sydney 2006, AUSTRALIA

email:laurent@maths.usyd.edu.au