MODIFIED ANALYTIC TRIVIALIZATION FOR

WEIGHTED HOMOGENEOUS FUNCTION-GERMS

Toshizumi FUKUI and Laurentiu PAUNESCU

A fundamental problem in real singularities is the classification of the real analytic function-germs. If one classifies them by the differentiable equivalence, the modulus may appear as the Whitney’s example (1.1) shows, thus we consider this equivalence relation too strong for the classification purpose. The equivalence relation by homeomorphisms does not cause modulus, but we do not expect a workable theory for this equivalence relation. For instance, the set of t = (t_{1}, t_{2}, t_{3}) so that f_{t}(x, y) = x^{5}+t_{1}x+t_{2}x^{2}+t_{3}x^{3}+y^{2} is topologically equivalent to f_{0} is not locally closed.

T.-C.Kuo is the first mathematician who has considered homeomorphisms becoming bianalytic isomorphisms after some blowing up, and he showed some basic trivialization theorem in [9, 10], and a finite classification theorem in [8, 11] for the equivalence relation induced by such homeomorphisms. In this paper, we consider this kind of trivialization. In particular, we want to know more families admitting such trivialization under some reasonable assumptions.

Let x = (x_{1}, ..., x_{n}) be a system of coordinates of R^{n} at the origin 0, and let I = [a_{1}, b_{1}] × ⋯ × [a_{m}, b_{m}] be a cube in R^{m}. Let f_{t} : (R^{n}, 0) → (R, 0) be a real analytic family of real analytic function-germs. We set F (x; t) = f_{t}(x). We denote the Taylor expansion of f_{t}(x) at the origin by ∑_{ν} c_{ν}(t)x^{ν}. Let ω = (ω_{1}, ..., ω_{n}) be an n-tuple of positive integers. Setting H_{j}(x; t) = ∑_{ν} c_{ν}(t)x^{ν} where the sum is taken over ν with ⟨ω, ν⟩ = ω_{1}ν_{1} + ⋯ + ω_{n}ν_{n} = j, we can write F (x; t) = H_{d}(x; t) + H_{d+1}(x; t) + ⋯, H_{d}(x; t) ≢ 0. We call H_{d} the weighted initial form of f_{t} about the weight ω. If ω = (1, ..., 1), this H_{d} is the usual initial form of f_{t}. We now quote the first trivialization theorem obtained by T.-C.Kuo in [9].

Theorem (0.1). Let f_{t} : (R^{n}, 0) → (R, 0) (t ∈ I) be a real analytic family of real analytic function-germs. If the initial forms of f_{t} (t ∈ I) define an isolated singularity at the origin, then f_{t} admits a MAT via the blowing up of R^{n} at the origin, that is, f_{t} is trivialized by a family of homeomorphisms coming from a family of bianalytic isomorphisms of the space obtained by the blowing up of R^{n} at the origin.

In this paper, we show the weighted version of this theorem. Namely, we have the following:

Theorem (0.2). Let f_{t} : (R^{n}, 0) → (R, 0) (t ∈ I) be a real analytic family of real analytic function-germs. If the weighted initial forms of f_{t} (t ∈ I) define an isolated singularity at the origin, then f_{t} admits a MAT via some toric modification of R^{n}, that is, f_{t} is trivialized by a family of homeomorphisms coming from a family of bianalytic isomorphisms of the space obtained by some toric modification of R^{n}.

For the definitions of the notions involved in the above theorems we send the reader to §1 and §4.

Speaking about topological trivialization only, many authors showed such trivialization theorem. For example, J.Damon [3] showed a topological trivialization theorem for A-finite weighted homogeneous map-germs. But his discussion does not apply to our situation, because we cannot use the partition of unity.

The authors would like to thank T.-C.Kuo and A.Parusinski for some discussions on the subject and the first named author for the hospitality during his stay at the University of Sydney. This project was partially supported by the ARC Grant A69530267

§1. Modified analytic trivialization. The motivation of modified analytic trivialization is explained by the following example.

Example (1.1)(H.Whitney [14]). Let W_{t} : (R^{2}, 0) → (R, 0) (t ∈ I) be a homogeneous polynomial-germ defined by W_{t}(x, y) = xy(x − y)(x − ty) for I = (1, ∞). Then, W_{t} (t ∈ I) is a topological trivial family that is not C^{1}-trivial.

Here, we are interested in the following observation due to T.-C.Kuo in [9]. Let π : X → R^{2} be the blowing up at the origin. Then the family W_{t} ∘π : (X, π^{−1}(0)) → (R, 0) is real analytically trivial along I. Since we may find a modulus situation like (1.1) after some blowing up, we should consider such a trivialization using at least compositions of blowing-ups or more widely proper analytic modifications. Thus, we lead to the definition of modified analytic trivialization.

Definition (1.2). Let π : X → R^{n} be a proper analytic modification. We say that f_{t} (t ∈ I) admits a modified analytic trivialization (MAT, for short) via π along I if there exists a t-level preserving real analytic isomorphism between two neighborhoods of π^{−1}(0) × I which induces a t-level preserving homeomorphism h between two neighbourhoods of 0 × I so that F ∘ h(x; t) is independent of t.

In [6, 7, 10, 13], there are several theorems that establish MAT.

§2. Review of homogeneous case. Because of its importance, we, very briefly, review the proof of (0.1) in [9] . Let π : X → R^{n} denote the blowing up at the origin. We first assume that m = 1 and set t = t_{1}. We consider the vector field V defined by

V = − ∑_{i=1}^{n} \frac{Q_{i}}{P} \frac{∂}{∂x_{i}} + \frac{∂}{∂t}, where P = ∑_{i=1}^{n} (\frac{∂F}{∂x_{i}} )^{2} , and Q_{i} = \frac{∂F}{∂t} \frac{∂F}{∂x_{i}} .

We see V is tangent to each level of F, whenever V is defined. Remember that X is the analytic submanifold of R^{n} × P^{n−1}(R) defined by x_{i}ξ_{j} = ξ_{i}x_{j} (1 ≤ i < j ≤ n) where P^{n−1}(R) is the (n − 1)-dimensional real projective space with homogeneous coordinate ξ = [ξ_{1} : ⋯: ξ_{n}]. On the subset of X defined by ξ_{s} ≠ 0, we have a coordinate system y = (y_{1}, ..., y_{n}) defined by y_{s} = x_{s}, y_{i} = ξ_{i}/ξ_{s}, i ≠ s, and π is expressed as x_{s} = y_{s}, x_{i} = y_{i}y_{s}, i ≠ s. Then for ξ_{s} ≠ 0,

\frac{∂}{∂x_{s}} = \frac{∂}{∂y_{s}} − ∑_{j≠s} \frac{y_{j}}{y_{s}} \frac{∂}{∂y_{j}}; \frac{∂}{∂x_{j}} = \frac{1}{y_{s}} \frac{∂}{∂y_{j}} , j ≠ s.

Since \frac{∂F}{∂t}, \frac{∂F}{∂x_{i}} are order d and d − 1 respectively, they are thus divisible by y_{s}^{d} and y_{s}^{d−1} respectively. We set P = y_{s}^{2d−2}P̃ and Q_{i} = y_{s}^{2d−1}Q̃_{i}. Then P̃ and Q̃_{i} are analytic. Since we can prove that P̃ is strictly positive near π^{−1}(0) × I, d(π^{−1} × id)V is analytically extendible there. The trajectory of this vector field yields the desired analytic isomorphism.

We next see the case m > 1. Let V_{j} be the vector field defined by

V_{j} = − ∑_{i=1}^{n} \frac{Q_{i}}{P} \frac{∂}{∂x_{i}} + \frac{∂}{∂t_{j}} , where P = ∑_{i=1}^{n} (\frac{∂F}{∂x_{i}} )^{2} , and Q_{i} = \frac{∂F}{∂t_{j}} \frac{∂F}{∂x_{i}} .

Then the same proof as above shows that it admits an analytic lift by π ×id. We denote its trajectory by H_{j}. If we define

H(y; t) = H_{m}(t_{m} − a_{m}; H_{m−1}(⋯; H_{1}(t_{1} − a_{1}; y, a)))

for t = (t_{1}, ..., t_{m}) ∈ I, and a = (a_{1}, ..., a_{m}), we obtain the desired analytic isomorphism.

§3. A vector field for the weighted homogeneous case. We consider the weighted version of the proof above. By the discussion above, it is enough to show the case m = 1. Thus, throughout the rest of this paper, we assume that m = 1 and set t = t_{1}. Unfortunately, the vector field V above does not have an analytic lift by the blowing up at the origin or by a toric modification (that will be defined in §4) under the assumption in (0.2). Thus, we should consider some another vector field.

Let F (x; t) = ∑_{ν} c_{ν}(t)x^{ν} be an analytic family of real analytic function-germs. We here consider the vector field V defined by

V = − ∑_{i=1}^{n} \frac{Q_{i}}{P} \frac{∂}{∂x_{i}} + \frac{∂}{∂t}, where P = ∑_{i=1}^{n} (\frac{∂F}{∂x_{i}} )^{2k_{i}s} , Q_{i} = \frac{∂F}{∂t} (\frac{∂F}{∂x_{i}} )^{2k_{i}s−1} ,

k_{i} = k/(d − ω_{i}), k = ∏_{i=1}^{n}(d − ω_{i}) and s is a positive integer. It is not hard to see this vector field V is tangent to each level of F, whenever V is defined. We also remark

the weighted initial form of P, about this weight, is ∑_{i=1}^{n} (\frac{∂H_{d}}{∂x_{i}} )^{2k_{i}s}.

§4. Toric modification. Let us recall some basic definitions and notation of toric modifications and fans. See [4, 5, 12], for a detailed discussion. Let M be the lattice Z^{n} and N its dual lattice Hom_{Z}(M, Z). We write an element of M as a column vector and an element of N as a row vector. Let e_{i} denote the i-th unit column vector (0, .., 0, 1, 0, ..., 0) and e^{i} the i-th unit row vector ^{t}(0, .., 0, 1, 0, ..., 0). We set M_{R} = M ⊗_{Z} R, and N_{R} = N ⊗_{Z} R, and denote ⟨ , ⟩ : N_{R} × M_{R} → R the canonical bilinear map defined by ⟨e^{i}, e_{j}⟩ = δ_{ij} where δ_{ij} = 1, if i = j; 0, otherwise. Let v_{1}, ..., v_{s} be vectors in N (resp. M). We set

σ = {r_{1}v_{1} + ⋯ + r_{s}v_{s} ∈ N_{R}(resp.M_{R}) : r_{i} ≥ 0, i = 1, ..., s}

and call it the cone generated by v_{1}, ..., v_{s}. We simply call a cone in N (resp. M) such a set. For a cone σ in N, set σ^{∨} = {u ∈ M_{R} : ⟨u, v⟩ ≥ 0, ∀v ∈ σ}. It is easy to see that σ^{∨} is a cone in M. Let R[M ∩ σ^{∨}] denote the algebra generated by the additive semi-group M ∩ σ^{∨} over the real number field R, and U_{σ} the set of R-valued points of Spec(R[M ∩ σ^{∨}]). If a cone σ_{1} in N is a face of another cone σ_{2} in N, then the natural inclusion σ_{2}^{∨} ⊂ σ_{1}^{∨} induces an open embedding U_{σ_{1}} ⊂ U_{σ_{2}}, since R[M ∩σ_{1}^{∨}] and R[M ∩ σ_{2}^{∨}] have a same quotient field and hence have a same generic point. Therefore, if Σ is a fan, that is, a finite collection of cones in N that forms a complex, we can glue U_{σ}’s (σ ∈ Σ) together and we obtain a (real) toric variety denoted by X_{Σ}. We say a cone σ in N is regular if σ is generated by a part of a basis of N. We say a fan is regular if σ is regular for each σ ∈ Σ. If Σ is a regular fan, then X_{Σ} is nonsingular.

Example (4.1). Let σ be the positive orthant in N_{R}. Then σ^{∨} is the positive orthant in M_{R} and R[M ∩ σ^{∨}] is the polynomial ring R[x_{1}, ..., x_{n}]. The set of faces of σ is a regular fan and the corresponding toric variety is R^{n}.

Let Σ be a fan that forms a subdivision of the positive orthant, that is, |Σ| := ⋃_{σ∈Σ} σ is the positive orthant in N_{R}. Then the natural embedding of the positive orthant in σ^{∨} (σ ∈ Σ) induces an embedding R[x_{1}, ..., x_{n}] ⊂ R[M ∩ σ^{∨}]. This defines a proper modification π = π_{Σ} : X_{Σ} → R^{n}. We call this a toric modification of R^{n}.

Assume that Σ is a regular fan that is a subdivision of the positive orthant. Then, for each n-cone σ in Σ, U_{σ} is isomorphic to R^{n} with some coordinate system (y_{1}, ..., y_{n}),

and the map π|_{U_{σ}} : U_{σ} → R^{n} can be expressed by x_{i} = ∏_{j=1}^{n} y_{j}^{a_{i}^{j}} for some integral n by n matrix (a_{i}^{j}) whose determinant is ±1. By definition, a^{j} = ^{t}(a_{1}^{j}, ..., a_{n}^{j}) (j = 1, ..., n) generate the cone σ.

Let Δ be a convex polyhedron in M_{R}, which coincides the positive orthant outside some compact set. We set σ_{F} = ⋃_{r≥0} r · (Δ − m), where m is a point in the relative interior of F, for a face F of Δ. Then the system Σ_{Δ} = {σ_{F} : F a face of Δ} forms a fan that is a subdivision of the first orthant. Then we obtain a proper modification X_{Σ_{Δ}} → R^{n}, and denote it by π_{Δ}. Then we have that π_{Δ} is isomorphic except the origin, that is, π|_{X−π_{Δ}^{−1}(0)} : X − π_{Δ}^{−1}(0) → R^{n} − 0 is an isomorphism.

§5. Analytic lift of the vector field. The purpose of this section is to show that the vector field V defined in §3 has an analytic lift by some toric modification under the supposition in (0.2). Set

Δ_{0} = {ν = (ν_{1}, ..., ν_{n}) ∈ M_{R} : ⟨ω, ν⟩ ≥ d, ν_{i} ≥ 0, i = 1, ..., n}.

Suppose Σ is a regular fan that is a subdivision of Σ_{Δ_{0}}. For a = ^{t}(a_{1}, ..., a_{n}) ∈ N, we set

ℓ(a) = min{⟨a, ν⟩ : c_{ν} ≠ 0},
ℓ′(a) = min\{⟨a, ν⟩ : \frac{dc_{ν}}{dt} ≠ 0\},
ℓ^{(i)}(a) = min{⟨a, ν⟩ : c_{ν} ≠ 0, ν_{i} ≥ 1}.

Obviously ℓ, ℓ′, ℓ^{(i)} are convex, and ℓ′(a) ≥ ℓ(a) and ℓ^{(i)}(a) ≥ ℓ(a).

Let σ is a regular cone in Σ, and a^{1}, ..., a^{n} be a system of generators for it. We consider the pullback of V on U_{σ} ×I with the system of coordinates y = (y_{1}, ..., y_{n}) and t. For shortness, we write a_{i} = (a_{i}^{1}, ..., a_{i}^{n}), ℓ = (ℓ(a^{1}), ..., ℓ(a^{n})), ℓ′ = (ℓ′(a^{1}), ..., ℓ′(a^{n})), and ℓ^{(i)} = (ℓ^{(i)}(a^{1}), ..., ℓ^{(i)}(a^{n})). Then, we can write

\frac{∂F}{∂x_{i}} = y^{ℓ^{(i)}−a_{i}} F̃_{i}, \frac{∂F}{∂t} = y^{ℓ′} F̃_{t}

for some analytic functions F̃_{i}, F̃_{t} in (y, t) near π^{−1}(0) × I. Thus,

P = ∑_{i=1}^{n} (\frac{∂F}{∂x_{i}} )^{2k_{i}s} = ∑_{i=1}^{n} y^{2k_{i}s(ℓ^{(i)}−a_{i})}F̃_{i}^{2k_{i}s}.

Since the weighted initial form of f_{t} defines an isolated singularity at the origin, the coefficient of x_{i}^{2ks/ω_{i}} in the Taylor expansion of P at 0 is not zero. Here we set

m(a) = min\{\frac{a_{1}}{ω_{1}} , ..., \frac{a_{n}}{ω_{n}} \}, R(a) = \{i : \frac{a_{i}}{ω_{i}} = m(a)\},

and m = (m(a^{1}), ..., m(a^{n})). Then we obtain P = y^{2mks}P̃ for some analytic function P̃ in (y, t) near π^{−1}(0) × I. We also have

m(a^{j}) = \frac{1}{k} min_{i}\{k_{i}(ℓ^{(i)}(a^{j}) − a_{i}^{j})\} = min_{i}\{\frac{ℓ^{(i)}(a^{j}) − a_{i}^{j}}{d − ω_{i}} \},

because the weighted initial form of P is ∑_{i=1}^{n} (\frac{∂H_{d}}{∂x_{i}} )^{2k_{i}s} which has non-zero coefficient

at the term x_{i}^{2ks/ω_{i}}.

Lemma (5.1). P̃ is everywhere positive near π^{−1}(0) × I.

Now we set

γ^{(i)}(a) = {ν ∈ M : ⟨a, ν⟩ = ℓ^{(i)}(a), c_{ν} ≠ 0, ν_{i} ≥ 1}, and
γ^{[i]}(a) = {ν ∈ M : ⟨a, ν⟩ = a_{i} + m(a)(d − ω_{i}), c_{ν} ≠ 0, ν_{i} ≥ 1}.

Lemma (5.2). Let Λ be a subset of {1, ..., n}, and a^{j} (j ∈ Λ) vectors with ⋂_{j∈Λ} R(a^{j}) ≠ ∅, If we set a = ∑_{j∈Λ} c_{j}a^{j} for positive numbers c_{j} (j ∈ Λ), then we have γ^{[i]}(a) = ⋂_{j∈Λ} γ^{[i]}(a^{j}).

Proof. ⊃ is obvious. We see the converse. For any ν ∈ γ^{[i]}(a),

⟨a, ν⟩ − ∑_{j∈Λ} c_{j}a_{i}^{j} = ∑_{j∈Λ} c_{j}m(a)(d − ω_{i}).

Since ⟨a^{j}, ν⟩ − a_{i}^{j} ≥ m(a^{j})(d − ω_{i}) for ν with c_{ν} ≠ 0, we have ν ∈ γ^{[i]}(a^{j}) for j ∈ Λ. □

Proof of (5.1). Let (y, t) be a point in π^{−1}(0) × I. We assume that y = (y_{1}, ..., y_{n}) satisfies y_{j} = 0, j ∈ Λ; y_{j} ≠ 0, j ∉ Λ, for some subset Λ of {1, ..., n}. It is enough to see P̃ is positive at (y, t). By definition of P̃, we have

P̃ = ∑_{i=1}^{n} ∏_{j=1^{n}} y_{j}^{2ks\{^{\frac{ℓ^{(i)(a^{j})}−a{j \atop i}}{d−ω_{i}}} −m(a^{j})\}}F̃_{i}^{2k_{i}s}.

Since y_{j} = 0 for j ∈ Λ, our assertion is equivalent to

\{y : ∑_{ν∈A} ν_{i}c_{ν} ∏_{j∉Λ} y_{j}^{⟨a^{j},ν⟩−ℓ^{(i)}(a^{j})} = 0 for i with \frac{ℓ^{(i)}(a^{j}) − a_{i}^{j}}{d − ω_{i}} = m(a^{j}), j ∈ Λ\}

is a subset of {∏_{j∉Λ} y_{j} = 0}, where A = ⋂_{j∈Λ} γ^{(i)}(a^{j}). Here, we set ỹ = (ỹ_{1}, ..., ỹ_{n}) where ỹ_{j} = y_{j}, if j ∉ Λ; 1 if j ∈ Λ. Then this is equivalent to

\{ỹ: ∑_{ν∈A} ν_{i}c_{ν} ∏_{j=1}^{n} ỹ_{j}^{⟨a^{j},ν⟩−ℓ^{(i)}(a^{j})} = 0 for i with \frac{ℓ^{(i)}(a^{j}) − a_{i}^{j}}{d − ω_{i}} = m(a^{j}), j ∈ Λ\}

is a subset of {∏_{j=1}^{n} ỹ_{j} = 0}. By setting x̃ = (x̃_{1}, ..., x̃_{n}) where x̃_{i} = ∏_{j=1}^{n} ỹ_{j}^{a_{i}^{j}} =

∏_{j∉Λ} y_{j}^{a_{i}^{j}}, this comes from

\{x̃ : ∑_{ν∈A} ν_{i}c_{ν}x̃^{ν} = 0 for i with \frac{ℓ^{(i)}(a^{j}) − a_{i}^{j}}{d − ω_{i}} = m(a^{j}), j ∈ Λ\}

is a subset of {∏_{i=1}^{n} x̃_{i} = 0}. Now we set B = ⋂_{j∈Λ} γ^{[i]}(a^{j}). Since γ^{[i]}(a^{j}) ≠ ∅ if and only if ℓ^{(i)}(a^{j}) = m(a^{j})(d − ω_{i}) + a_{i}^{j}, our assertion is equivalent to

(5.3)  \{x : \frac{∂}{∂x_{i}} (∑_{ν∈B} c_{ν}x^{ν}) = 0 : i = 1, ..., n\} ⊂ \{x_{1} ⋯ x_{n} = 0\}.

By (5.2), if we set a = ∑_{j∈Λ} c_{j}a^{j} for c_{j} > 0, we have B = γ^{[i]}(a).

We set a = \overline{c}_{0}ω + ∑_{j=1}^{n} \overline{c}_{j}e^{j} (\overline{c}_{j} ≥ 0, \overline{c}_{0} > 0, \overline{c}_{1} ⋯ \overline{c}_{n} = 0) and E(a) = \{j : \overline{c}_{j} > 0\}. Then we have

γ^{[i]}(a) = γ^{[i]}(ω) ∩ ⋂_{j∈E(a)} γ^{[i]}(e^{j}) (by (5.2))
= γ^{[i]}(ω) ∩ {ν : ν_{j} = 0 for j ∈ E(a) − {i}, ν_{i} = 1 if i ∈ E(a)}.

Thus (5.3) is equivalent to

{gradH_{d}(x; t) = 0} ∩ {x_{j} = 0, j ∈ C} ⊂ {∏_{j∉C} x_{j} = 0},

where C = ⋃_{j∈Λ} E(a^{j}). This always holds, since H_{d} defines an isolated singularity at the origin. This completes the proof of (5.1). □

We next write down Q_{i} as a function in variables y and t. By definition,

Q_{i} = \frac{∂F}{∂t} (\frac{∂F}{∂x_{i}} )^{2k_{i}s−1} = y^{ℓ′+(2k_{i}s−1)(ℓ^{(i)}−a_{i})}F̃_{t}F̃_{i}^{2k_{i}s−1}.

Therefore the exponent of y in Q_{i}/P is ℓ′ + (2k_{i}s − 1)(ℓ^{(i)} − a_{i}) − 2mks, which is equal to

(5.4)  (ℓ′ − ℓ) + (ℓ − ℓ^{(i)}) + 2sk \{\frac{ℓ^{(i)} − a_{i}}{d − ω_{i}} − m\} + a_{i}.

We remark that each term of a component of (5.4) is not negative except the second. We say that a = ^{t}(a_{1}, ..., a_{n}) ∈ N is i-good if ℓ^{(i)}(a) > a_{i}+m(a)(d−ω_{i}) or ℓ(a) = ℓ^{(i)}(a).

Lemma (5.5). If a = ^{t}(a_{1}, ..., a_{n}) is i-good and s is large enough, then we have

(ℓ′(a) − ℓ(a)) + (ℓ(a) − ℓ^{(i)}(a)) + 2sk \{\frac{ℓ^{(i)}(a) − a_{i}}{d − ω_{i}} − m(a)\} ≥ 0.

Proof. Obvious. □

Lemma (5.6). If the map π|_{U_{σ}} : U_{σ} → R^{n} is expressed by

x_{i} = y_{i}y_{k+1}^{a_{i}^{k+1}} ⋯ y_{n}^{a_{i}^{n}}, i = 1, ..., k; x_{i} = y_{k+1}^{a_{i}^{k+1}} ⋯ y_{n}^{a_{i}^{n}}, i = k + 1, ..., n,

then y_{k+1}^{a_{i}^{k+1}} ⋯ y_{n}^{a_{i}^{n}} \frac{∂}{∂x_{i}} has an analytic lift on U_{σ}.

Proof. By elementary calculation, we have ∑_{i=1}^{n} a_{i}^{j}x_{i} \frac{∂}{∂x_{i}} = y_{j} \frac{∂}{∂y_{j}}. Thus, we obtain

x_{i} \frac{∂}{∂x_{i}} = y_{i} \frac{∂}{∂y_{i}} , i = 1, ..., k; x_{i} \frac{∂}{∂x_{i}} = ∑_{j=k+1}^{n} b_{i}^{j}y_{j} \frac{∂}{∂y_{j}} , i = k + 1, ..., n,

where (b_{i}^{j})_{i,j=k+1,...,n} is the inverse matrix of (a_{i}^{j})_{i,j=k+1,...,n}. This implies the lemma. □

We say that a cone is good if each its 1-dimensional face is generated by a vector that is either an i-good vector for each i, or an e^{i}. We say a fan is good if each cone in it is good. Therefore, if there is a good subdivision Σ of Σ_{Δ_{0}} and s is sufficiently large, V has an analytic lift by the toric modification π corresponding to Σ, because of (5.1), (5.4), (5.5) and (5.6). By the proof of (5.6), this analytic lift is tangent to each irreducible component of π^{−1}(0) × I. If the toric modification π is isomorphic except the origin, the trajectory of this vector field defines a t-level preserving homeomorphism of R^{n} × I near 0 × I, and this completes the proof. In the next section, we will show the existence of such a good subdivision.

§6. Good subdivision. In this section, we show the following:

Proposition (6.1). There is a good subdivision of Σ_{Δ_{0}} so that the corresponding toric modification is isomorphic except the origin.

Lemma (6.2). Let k be a positive integer with k < n, a^{k+1}, ..., a^{n} i-good vectors and suppose that ⋂_{j=1}^{k} R(e^{j}) ∩ ⋂_{j=k+1}^{n} R(a^{j}) ≠ ∅. Let a = ∑_{j=1}^{k} c_{j}e^{j} + ∑_{j=k+1}^{n} c_{j}a^{j} be an integral vector with 0 ≤ c_{i} < 1. Then a is i-good.

Proof. In general, we have ℓ^{(i)}(a) ≥ ∑_{j=1}^{k} c_{j}ℓ^{(i)}(e^{j}) + ∑_{j=k+1}^{n} c_{j}ℓ^{(i)}(a^{j}). If ℓ^{(i)}(a) > (d − ω_{i})m(a) + a_{i}, (6.2) is obvious. We assume ℓ^{(i)}(a) = (d − ω_{i})m(a) + a_{i}. Then it follows that

ℓ^{(i)}(a) = ∑_{j=1}^{k} c_{j}ℓ^{(i)}(e^{j}) + ∑_{j=k+1}^{n} c_{j}ℓ^{(i)}(a^{j}),

and ℓ^{(i)}(e^{j}) = δ_{ij}, j = 1, ..., k; ℓ^{(i)}(a^{j}) = ℓ(a^{j}) = (d − ω_{i})m(a^{j}) + a_{i}^{j}, j = k + 1, ..., n. So

ℓ^{(i)}(a) = ∑_{j=1}^{k} c_{j}ℓ^{(i)}(e^{j}) + ∑_{j=k+1}^{n} c_{j}ℓ^{(i)}(a^{j}) = c_{i} + ∑_{j=k+1}^{n} c_{j}ℓ^{(i)}(a^{j}).

Thus we obtain

ℓ(a) ≥ ∑_{j=1}^{n} c_{j}ℓ(e^{j}) + ∑_{j=k+1}^{n} ℓ(a^{j})

= ∑_{j=1}^{n} c_{j}ℓ(e^{j}) + ∑_{j=k+1}^{n} ℓ^{(i)}(a^{j})

= ∑_{j=1}^{n} c_{j}ℓ(e^{j}) + ℓ^{(i)}(a) − c_{i}.

Since the weighted initial form of f_{t} defines an isolated singularity at the origin, ℓ(e^{j}) can be 0 or 1, and only one can be 1. We thus obtain 0 ≤ ℓ^{(i)}(a) − ℓ(a) < 1, which implies ℓ^{(i)}(a) = ℓ(a). □

Lemma (6.3). Let k be a positive integer with k < n, a^{k+1}, ..., a^{n} vectors generating good cones and suppose that ⋂_{j=1}^{k} R(e^{j}) ∩ ⋂_{j=k+1}^{n} R(a^{j}) ≠ ∅. Then there is a good subdivision for the cone generated by e^{1}, ..., e^{k}, a^{k+1}, ..., a^{n}.

Proof. We may assume that this cone is not regular. Then the volume v of the parallelogram generated by e^{s} (s = 1, ..., k) and a^{t} (t = k + 1, .., n) is not 1. We also have an integral vector a = ∑_{j=1}^{k} c_{j}e^{j} + ∑_{j=k+1}^{n} c_{j}a^{j} with 0 ≤ c_{j} < 1, ∑_{j=k+1}^{n} c_{j} > 0. According to (6.2), we have that a is i-good. Then a divides in n-cones, and the parallelogram generated by a, e^{s} (s = 1, ..., k) and a^{t} (t = k + 1, .., n) except e^{j} or a^{j} has volume c_{j}v, and less than v. Thus, we can show (6.3) by induction on v. □

By the proof of (6.3), it is easy to see (6.1). □

§7. Examples.

Example (7.1)(J.Briançon - J.P.Speder [2]). Let f_{t} : (R^{3}, 0) → (R, 0) ( t ≠ t_{0} := −15^{1/7}(7/2)^{4/5}/3) be the family defined by f_{t}(x, y, z) = z^{5} + ty^{6}z + xy^{7} + x^{15}. This is a weighted homogeneous polynomial with weight (1, 2, 3) and weighted degree 15, and defines isolated singularity at the origin, whenever t ≠ t_{0}. Thus this admits a MAT along I, where I is an interval not containing t_{0}. This fact was already remarked in [7].

Example (7.2). Let f_{t} : (R^{3}, 0) → (R, 0) ( t ∈ I = (−∞, ∞)) be the family of function-germs defined by f_{t}(x, y, z) = x^{15} + y^{8}x + z^{5} + tzy^{7}. This is a weighted homogeneous polynomial with weight (4, 7, 12) and weighted degree 60 if t = 0. Since this family satisfies the assumption of (0.2), it admits MAT along I.

§8. Epilogue. We set K = R or C. Let f : (K^{n}, 0) → (K, 0) be an analytic functiongerm. We say that a proper analytic modification π : X → K^{n} gives a resolution of f if for each point x ∈ X there is a system of coordinates at x so that f ∘π is expressed by a monomial in this coordinates. Let f_{t} : (K^{n}, 0) → (K, 0) (t ∈ I) be an analytic family of analytic function-germs. We set F (x; t) = f_{t}(x). We say a proper analytic modification π : X → K^{n} gives a simultaneous resolution of f_{t} (t ∈ I) if for each point x ∈ X there is a family of systems of coordinates at x depending on t ∈ I so that F ∘ (π × id) is expressed by a monomial in this coordinates.

Let f : (R^{n}, 0) → (R, 0) be a real analytic function-germ. Suppose that the weighted initial form of f, with respect to the weight ω (and say of degree d), defines isolated singularity at the origin. Then there is a weighted homogeneous polynomial g with weight ω and weighted degree d and satisfying the following conditions:

(1) g defines an isolated singularity at the origin.

(2) g is non-degenerate. (See [1, 6] e.g., for its definition.)

(3) there is an analytic family f_{t} (t ∈ [0, 1]) so that f_{0} = f and f_{1} = g, and that its weighted initial form H_{d}(x; t) defines an isolated singularity at the origin for each t ∈ [0, 1].

Then, by (0.2), this family f_{t} admits a MAT, and thus admits a simultaneous resolution, because this toric modification gives a resolution of g. In particular, this toric modification gives a resolution of f.

Let f_{t} : (C^{n}, 0) → (C, 0) (t ∈ I) be a real analytic family of holomorphic functiongerms. Suppose that the weighted initial form H_{d} of f_{t}, about the weight ω, defines isolated singularity at the origin. We set F (x; t) = f_{t}(x). Considering the vector field V defined by

V = − ∑_{i=1}^{n} \frac{Q_{i}}{P} \frac{∂}{∂x_{i}} + \frac{∂}{∂t}, where P = ∑_{i=1}^{n} |\frac{∂F}{∂x_{i}} |^{2k_{i}s} , Q_{i} = \frac{∂F}{∂t} |\frac{∂F}{∂x_{i}} |^{2k_{i}s−2} \overline{\frac{∂F}{∂x_{i}}} ,

we can construct a t-level preserving homeomorphism of C^{n} × I near 0 × I coming from a real analytic isomorphism X × I near π^{−1}(0) × I where π : X → C^{n} is a (complex) toric modification constructed by the way similar to the above. Here, \overline{F} denotes the complex conjugation of F.

Thus, we obtain the following facts, which do not seem to be appeared in literature.

(1) Let f : (K^{n}, 0) → (K, 0) be an analytic function-germ. If the weighted initial form of f about the weight ω defines an isolated singularity at the origin, then a toric modification corresponding to a good subdivision of Σ_{Δ_{0}} gives a resolution of f.

(2) Let f_{t} : (K^{n}, 0) → (K, 0) (t ∈ I) be an analytic family of analytic functiongerms. If the weighted initial form of f_{t} (t ∈ I) about the weight ω defines an isolated singularity at the origin, then a toric modification corresponding to a good subdivision of Σ_{Δ_{0}} gives a simultaneous resolution of f_{t} (t ∈ I).

References

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[2] J.Briançon and J.Speder, La trivialté topolgique n’implque pas les conditions de Whitney, C.R. Acad. Sci., 280 (1975), Paris, 365–367.

[3] J.Damon, Finite determinacy and topological triviality I, Invent. math. 62 (1980), 299–324.

[4] V.I.Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97–154.

[5] W.Fulton, Introduction to Toric Varieties, Ann. of Math. Studies 131, Princeton Univ. Press, 1993.

[6] T.Fukui and E.Yoshinaga, The modified analytic trivialization of family of real analytic functions, Invent. math. 82 (1985), 467–477.

[7] T.Fukui, The modified analytic trivialization via weighted blowing up, J. Math. Soc. Japan. 44 (1992), 455–459.

[8] T.-C.Kuo, Une classification des singularité réeles, C.R. Acad. Sci., 288 (1979), Paris, 809–812.

[9] T.-C.Kuo, The modified analytic trivialization of singularities, J. Math. Soc. Japan. 32 (1980), 605–614.

[10] T.-C.Kuo and J.N.Ward, A theorem on almost analytic equisingularities, J. Math. Soc. Japan. 33 (1981), 471–484.

[11] T.-C.Kuo, On classification of real singularities, Invent. math. 82 (1985), 257–262.

[12] T.Oda,, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Band 14, Springer-Verlag 1987.

[13] E.Yoshinaga, The modified analytic trivialization of real analytic family via blowing-ups, J. Math. Soc. Japan 40 (1988), 161–179.

[14] H.Whitney, Local properties of analytic varieties, A Symposium in Honor of M.Morse, (Ed. by S.S.Cairns), Princeton Univ. Press., 1965, pp. 205–244.

Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubbo, Urawa, 338 Japan

School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

E-mail address: tfukui@rimath.saitama-u.ac.jp, paunescu l@maths.su.oz.au