Abstract. Let G : C

(1)

POLONICI MATHEMATICI LXIII.2 (1996)

Logarithmic structure of the generalized bifurcation set

by S. Janeczko (Warszawa)

Abstract. Let G : C

ⁿ

×C

^r

→ C be a holomorphic family of functions. If Λ ⊂ C

ⁿ

×C

^r

, π

r

: C

ⁿ

× C

^r

→ C

^r

is an analytic variety then

Q

Λ

(G) = {(x, u) ∈ C

ⁿ

× C

^r

: G(·, u) has a critical point in Λ ∩ π

r⁻¹

(u)}

is a natural generalization of the bifurcation variety of G. We investigate the local structure of Q

_Λ

(G) for locally trivial deformations of Λ

0

= π

_r⁻¹

(0). In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.

1. Introduction. Motivation of this paper lies in theoretical questions in optics where a central role is played by isotropic, Lagrangian and coisotropic varieties in a symplectic space. The geometrical framework convenient for investigations of these varieties is based mainly on the action of symplectic relations (cf. [4]).

Let Ω = (T

^∗

R

^k

× T

^∗

R

ⁿ

, π

^∗₂

ω

_Rⁿ

− π

^∗₁

ω

_R^k

) be a product symplectic space.

Lagrangian submanifolds of Ω (symplectic relations) act on subsets of (T

^∗

R

^k

, ω

_R^k

) preserving their symplectic properties. In this way one can investigate the symplectic projections π

_Rⁿ

|

_S

: S → R

ⁿ

using the representa- tion of S as the image under a symplectic relation L ⊂ Ω of a subset Λ of the zero-section of T

^∗

R

^k

, i.e.

S = L(Λ) = {p ∈ T

^∗

R

ⁿ

: ∃

p∈Λ¯

(p, p) ∈ L}.

For practical purposes one seeks to classify germs of the projections π

_Rⁿ

|

S

and describe the structure of the corresponding variety of critical values.

Assuming that L is generated by a smooth function G : R

^k

× R

ⁿ

→ R we easily find that this variety is defined as a generalized bifurcation diagram

Q

Λ

(G) = {q ∈ R

ⁿ

: G(·, q) has a critical point belonging to Λ}.

In this paper we study the generalized bifurcation varieties of complex analytic families G using the technical tools of the theory of singularities

1991 Mathematics Subject Classification: Primary 58C27, 58F14; Secondary 57R45, 53A04.

Key words and phrases: bifurcations, singularities, logarithmic stratifications.

[187]

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of functions on varieties (cf. [3]). In Section 2 we provide the classification scheme of such varieties and introduce the notion of logarithmic stratification. In Section 3 we adapt to our Λ-bifurcation varieties the method for construction of logarithmic vector fields which is well known for the standard bifurcation and discriminant varieties (cf. [2, 14]). The specific algorithm ex- plicitly calculating the tangent vector fields to Q

Λ

(G) and the representative examples of Λ-bifurcation varieties are discussed in Section 4.

2. Classification of generalized bifurcation varieties. Let O

n

be the ring of germs of holomorphic functions at 0 ∈ C

ⁿ

. Let (Λ, 0) ⊂ (C

ⁿ

, 0) be the germ of a reduced analytic subvariety of C

ⁿ

at 0:

Λ = {x ∈ C

ⁿ

: F (x) = 0}, F ∈ O

n

.

The group of germs of diffeomorphisms φ : (C

ⁿ

, 0) → (C

ⁿ

, 0) which preserve Λ is denoted by G

Λ

. If J

Λ

denotes the ideal in O

n

consisting of germs of functions vanishing on Λ, then for φ ∈ G

Λ

the induced isomorphism φ

^∗

: O

_n

→ O

_n

preserves J

Λ

.

Two function-germs g

1

, g

2

: (C

ⁿ

, 0) → (C, 0) are called G

Λ

-equivalent if there is a diffeomorphism φ ∈ G

Λ

with g

1

◦ φ = g

₂

[3, 10].

We obtain elements of G

Λ

by integrating vector fields tangent to Λ.

Definition 2.1. We denote by Ξ

Λ

the O

n

-module of logarithmic vector fields for Λ, i.e. holomorphic vector fields on (C

ⁿ

, 0), which, if considered as derivations, say v : O

n

→ O

_n

, satisfy

v.h ∈ J

Λ

for all h ∈ J

Λ

.

Modules of holomorphic vector fields of this type are discussed in [11].

A function-germ g : (C

ⁿ

, 0) → (C, 0) is k-G

Λ

-determined if for all e g : (C

ⁿ

, 0) → (C, 0) with the same k-jet as g the germs g and e g are G

_Λ

-equivalent. Given a germ h : (C

ⁿ

, 0) → (C, 0), a germ G : (C

ⁿ

×C

^r

, 0) → (C, 0) is called a deformation of h if G(x, 0) = h(x). Formally we look on a deformation of h as a pair (G, r). Given two deformations (H, r), (G, q) of h, a morphism (Φ, l) : H → G between them is defined as follows:

1. Φ : (C

ⁿ

× C

^r

, 0) → (C

ⁿ

× C

^q

, 0) has the form Φ(x, u) = (φ(x, u), u) with φ(·, 0) = id

_Cⁿ

and φ(·, u) ∈ G

Λ

for all u near 0 ∈ C

^r

.

2. l : (C

^r

, 0) → (C

^q

, 0) is such that

G(φ(x, u), l(u)) = H(x, u).

A deformation (G, q) of h is G

Λ

-versal if for any unfolding (H, r) of h there is a morphism (Φ, l) : H → G.

Two deformations of h are equivalent if there exists a morphism between

them which is an isomorphism.

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Let U ⊂ (C

ⁿ

, 0) be an open, sufficiently small subset of C

ⁿ

. We can also consider the sheaf O

U

of holomorphic functions on U , and the sheaf Der

U

of holomorphic vector fields on U , together with its subsheaf Ξ

Λ

. Following [11] we introduce the logarithmic stratification of U determined by Ξ

Λ

.

Definition 2.2. Let {Λ

α

: α ∈ I} be a stratification of U with the following properties:

1. Each stratum Λ

α

is a smooth connected immersed submanifold of U and U = S

α∈I

Λ

α

.

2. If x ∈ Λ

α

then T

x

Λ

α

coincides with Ξ

Λ

(x).

3. If Λ

α

, Λ

β

are two distinct strata with Λ

α

meeting the closure Λ

β

of Λ

β

, then Λ

α

is contained in the boundary ∂Λ

β

of Λ

β

.

Then {Λ

α

: α ∈ I} is called a logarithmic stratification of Λ and Λ

α

is a logarithmic stratum.

For any variety Λ and sufficiently small U there always exists a unique logarithmic stratification of U .

The aim of this note is to construct the logarithmic stratification for generalized bifurcation varieties, and so to construct an appropriate module of logarithmic vector fields Ξ

Λ

.

Let g : (C

ⁿ

, 0) → (C, 0), g ∈ O

n

. We define the Jacobi ideal of g by

∆

Λ

(g) = {v.g : v ∈ Ξ

Λ

}.

If ∆

Λ

(g) ⊃ m

^k_n

, then g is (k + 1)-G

Λ

-determined, i.e. for all g : (C e

ⁿ

, 0) → (C, 0) with the same (k + 1)-jet as g the germs g, e g are G

Λ

-equivalent. Here m

n

is the maximal ideal of O

n

. As in the usual singularity theory setting [1] a deformation (G, r) of g is G

Λ

-versal if and only if

∂G

∂u

1

(x, 0), . . . , ∂G

∂u

r

(x, 0) span O

n

/∆

Λ

(g).

We know (cf. [2]) that if the set-germ

{x ∈ C

ⁿ

: v.g(x) = 0 for all v ∈ Ξ

Λ

}

at 0 is {0} or empty then g has a G

Λ

-versal deformation. If the number µ = dim

_C

O

n

/∆

Λ

(g) is finite, then it is called the multiplicity of g on Λ at 0, and is also denoted by µ

Λ

(g).

Let (G, r) be a deformation of g.

Definition 2.3. The analytic variety

Q

Λ

(G) = {u ∈ C

^r

: G(·, u) has a critical point on Λ}

is called the Λ-bifurcation variety of the family G.

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Define

Σ

Λ

(G) =

(x, u) ∈ C

ⁿ

× C

^r

: ∂G

∂x

i

(x, u) = 0, F (x) = 0

, where Λ = F

⁻¹

(0), F ∈ O

n

. Then we see that

Q

Λ

(G) = π

r

(Σ

Λ

(G)), where π

r

: C

ⁿ

× C

^r

→ C

^r

.

Example 2.4. As a natural example we consider the simplest Λ-bifurcation varieties corresponding to singularities of functions on regular bound- aries (cf. [1]). Let Λ = {(y, x) ∈ C

ⁿ⁺¹

: y = 0}, x = (x

1

, . . . , x

n

). It is easy to check that for B

k

and C

k

singularities Q

Λ

(G) are smooth hypersurfaces.

For the F

4

singularity

G(y, x, u) = y

²

+ x

³

+ u

1

xy + u

2

y + u

3

x the Λ-bifurcation variety Q

Λ

(G) is the Whitney cross cap

3u

²

+ u

3

u

²₁

= 0.

By straightforward calculations we prove that for unimodal, corank one boundary singularities of smallest codimension µ = 6:

F

1,0

: G(y, x, u) = x

³

+ bx

²

y + y

³

+ u

1

xy

²

+ u

2

xy + u

3

y

²

+ u

4

x + u

5

y, K

4,2

: G(y, x, u) = x

⁴

+ ax

²

y + y

²

+ u

1

x

²

y + u

2

x

²

+ u

3

yx + u

4

x + u

5

y, the Λ-bifurcation varieties are:

1. The trivial extension of the Whitney cross cap variety in the case F

1,0

. 2. The generalized Whitney cross cap (cf. [1], Section 9.6), given in the following parametric form:

u

1

= s, u

2

= t, u

3

= w, u

4

= −4x

³

− 2tx, u

5

= −(a + s)x

²

− wx.

For simplest unimodal, corank two boundary singularity of type L

6

: G(y, x, u) = x

²₁

x

2

+ x

³₂

+ yx

1

+ ayx

2

+ u

1

yx

2

+ u

2

x

²₁

+ u

3

x

1

+ u

4

x + u

5

y, the Λ-bifurcation variety Q

Λ

(G) is parametrized in the form

u

1

= s, u

2

= t, u

3

= −2x

1

x

2

− 2x

₁

t, u

4

= −x

²₁

− 3x

²₂

, u

5

= −x

1

− sx

₂

− ax

₂

and is an opening of the Σ

²

-Boardmann singular mapping C

⁴

→ C

⁴

.

3. Logarithmic vector fields. We denote by Sing(Σ

Λ

(G)) the singular part of Σ

Λ

(G). Then Σ

Λ

(G) − Sing(Σ

Λ

(G)) decomposes into analytic strata Σ

_Λ^α

(G), α ∈ I. We consider the family of mappings π

_r^α

= π

r

|

_Σ^α

Λ(G)

. Critical points of these mappings are described by an extra n equations:

rank ∂

²

G/∂x

i

∂x

j

∂F/∂x

j

(x, u) < n.

(5)

We denote by Γ

_r^α

= Γ (π

_r^α

) the set of critical values of the mapping π

_r^α

. Now we assume that (G, r) is a G

Λ

-versal deformation of g. Let g

0

, . . . . . . , g

µ−1

be a basis of the quotient space O

n

/∆

Λ

(g) with g

0

= 1 and g

i

∈ m

n

. Then by the equivalence of deformations we get a miniver- sal deformation of g ∈ m

²_n

(with minimal number of deformation para- meters u), i.e.

G(x, u) =

µ−1

X

i=1

u

i

g

i

(x) + g(x).

Now we have the following

Proposition 3.1. If ξ ∈ Ξ

QΛ(G)

then ξ is π

r

-liftable, i.e. there exists a germ of a holomorphic vector field e ξ on C

ⁿ

× C

^r

which is tangent to Σ

Λ

(G) at 0 and

ξ ◦ π

r

= dπ

r

◦ e ξ.

P r o o f. We see that ξ lifts by π

r

at every point u ∈ C

^r

outside π

r

(Sing(Σ

Λ

(G))) ∪ S

α∈I

Γ

_r^α

to a holomorphic vector field e ξ

⁰

on C

ⁿ

× C

^r

tangent to Σ

Λ

(G) and defined off a set of codimension 2 in C

ⁿ

× C

^r

, namely

C

ⁿ

× π

_r

(Sing(Σ

Λ

(G))) ∪ [

α∈I

Γ

_r^α

.

By Hartog’s extension theorem [9], e ξ

⁰

extends to a holomorphic vector field ξ tangent to Σ e

Λ

(G).

Now following the methods introduced in [3, 14] we give an algorithm for construction of the module Ξ

QΛ(G)

of vector fields for versal G. This algorithm is a generalization of a similar one constructed in [7] for vector fields tangent to the usual bifurcation varieties.

By Proposition 3.1, to obtain elements of Ξ

QΛ(G)

we have to construct all π

r

-lowerable vector fields e ξ tangent to Σ

Λ

(G).

Now we define the ideal J

ΣΛ(G)

= ∂G

∂x

1

(x, u), . . . , ∂G

∂x

n

(x, u), F (x)

O

_n+r

. Then the germ of the vector field

ξ = e

n

X

i=1

β

i

∂

∂x

i

+

r

X

j=1

γ

j

∂

∂u

j

, β

i

, γ

j

∈ O

_n+r

, at 0 ∈ C

ⁿ

× C

^r

, which is tangent to Σ

Λ

(G), has the property

ξ e ∂G

∂x

i

(x, u)

∈ J

_Σ_Λ_(G)

, i = 1, . . . , n, (1)

ξ(F (x)) ∈ J e

ΣΛ(G)

.

(2)

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Lemma 3.2. Let ξ =

r

X

i=1

α

i

(u) ∂

∂u

i

, ξ ∈ Ξ

QΛ(G)

.

The vector field e ξ ∈ Ξ

ΣΛ(G)

is a lifting of ξ if and only if for some β

i

∈ O

n+r

and v

i

∈ Ξ

_Λ

, i = 1, . . . , n, we have (3)

n

X

j=1

β

j

v

j

∂G

∂x

i

(x, u)

+

µ−1

X

j=1

α

j

(u) ∂g

j

∂x

i

∈ J

_Σ_Λ_(G)

, where G is G

Λ

-versal ,

G(x, u) =

µ−1

X

i=1

u

i

g

i

(x) + g(x).

P r o o f. By straightforward check of the conditions (1) and (2).

Now we use the arguments working for the bifurcation and discriminant sets. Consider the ideal

∆ e

Λ

(G) = hv

i

.GiO

n+r

in O

n+r

, where v

i

are generators of Ξ

Λ

. Since G is G

Λ

-versal, by the preparation theorem the quotient module

A = O

n+r

/ e ∆

Λ

(G)

is a free O

r

-module generated by 1, g

1

, . . . , g

µ−1

. In fact, take π(x, u) → u, and look on A as an O

n+r

-module. Then A is a finite O

r

-module if and only if A/(π

^∗

m

r

)A is finite over C. We see that

A/(π

^∗

m

r

)A ∼ = O

n+r

/(hv

i

.Gi + m

r

O

_n+r

)

∼ = O

n

/hv

i

.G(x, 0)iO

n

∼ = {1, g

1

, . . . , g

µ−1

}

_C

. Thus for any h ∈ O

n+r

we can write

(4) h(x, u) =

n

X

i=1

β

i

(x, u)(v

i

.G)(x, u) +

µ−1

X

j=1

α

j

(u)g

j

(x) + α(u) for some β

i

∈ O

_n+r

, α

i

∈ O

_r

and α ∈ O

r

.

Now we have the basic result.

Theorem 3.3. Let h ∈ O

n+r

and suppose that

∂h

∂x

i

(x, u) ∈ J

_Σ_Λ_(G)

, i = 1, . . . , n.

Then the vector field

ξ =

r

X

i=1

α

i

(u) ∂

∂u

i

,

(7)

where α

i

, 1 ≤ i ≤ µ − 1, are defined in (4) and α

i

, i ≥ µ, are arbitrary holomorphic functions from O

r

, is tangent to the Λ-bifurcation variety of the family G.

P r o o f. Take h in the form (4). For derivatives of h we have

∂h

∂x

i

(x, u) =

n

X

j=1

∂β

j

∂x

i

(v

j

.G) +

n

X

j=1

β

j

∂

∂x

i

(v

j

.G) +

µ−1

X

j=1

α

j

(u) ∂g

j

∂x

i

(x) and by assumptions this belongs to J

ΣΛ(G)

. We also have

n

X

j=1

β

j

∂

∂x

i

(v

j

.G) =

n

X

j=1

β

j

v

j

∂G

∂x

i

mod(J

ΣΛ(G)

).

So by Lemma 3.2 we obtain the lifting formula (3) for the vector field ξ = P

r

i=1

α

i

∂/∂u

i

, which is tangent to Q

Λ

(G).

One can also obtain the converse, which results immediately from the proof of Theorem 3.3.

Corollary 3.4. Let ξ = P

r

i=1

α

i

(u)∂/∂u

i

be a tangent vector field to Q

Λ

(G). Then for some h ∈ O

n+r

,

(5) h =

n

X

i=1

β

i

(v

i

G) +

µ−1

X

j=1

α

j

g

j

+ α, where β

i

∈ O

_n+r

, α ∈ O

r

and ∂h/∂x

i

∈ J

_Σ_Λ_(G)

.

P r o o f. Take h in the form (5), where

n

X

i=1

β

i

v

i

+

µ−1

X

j=1

α

j

∂

∂u

j

∈ Ξ

_Σ_Λ_(G)

. Then by a simple check we find that ∂h/∂x

i

∈ J

_Σ_Λ_(G)

.

One can easily check that the space of germs h ∈ O

n+r

such that

∂h/∂x

i

(x, u) ∈ J

_Σ_Λ_(G)

, i = 1, . . . , n, is an O

r

-module, which we denote by H

G

.

4. An algorithm. Now we present an algorithm which is useful in ob- taining all tangent vector fields to Q

Λ

(G). We see that

hF iJ

_Σ_Λ_(G)

+ e ∆

²_Λ

(G) ⊂ H

G

.

Since ∆

Λ

(g) contains some power of the maximal ideal m

n

, also the space O

_n

∆

²_Λ

(g) + hF iJ

Λ

(g) , J

Λ

(g) = ∂g

∂x

1

, . . . , ∂g

∂x

n

, F (x)

,

is finite-dimensional with C-basis, say, {f

1

, . . . , f

N

}.

(8)

By the preparation theorem {f

i

}

^N_i=1

also generates O

n+r

∆ e

²_Λ

(G) + hF iJ

ΣΛ(G)

as an O

r

-module.

Now any element h ∈ H

G

can be written in the form h(x, u) =

N

X

i=1

φ

i

(u)f

i

(x) +

n

X

i,j=1

β

i,j

(x, u) ∂G

∂x

i

(x, u) ∂G

∂x

j

(x, u)

+

n

X

i=1

γ

i

(x, u) ∂G

∂x

i

(x, u)F (x) + γ

0

(x, u)F (x)

²

, where β

i,j

, γ

i

, γ

0

∈ O

_n+r

and we seek elements φ

i

∈ O

_r

such that

N

X

i=1

φ

i

(u) ∂f

i

∂x

j

∈ J

_Σ_Λ_(G)

, 1 ≤ j ≤ n.

We show how to work with this approach and algorithm in several concrete cases.

4.1. Let Λ = {(y, x) ∈ C

ⁿ⁺¹

: y = 0}, x = (x

1

, . . . , x

n

). Then for some g ∈ O

n+1

and the versal unfolding G of g we have

∆

Λ

(g) =

y ∂g

∂y , ∂g

∂x

1

, . . . , ∂g

∂x

n

O

n+1

,

∆ e

Λ

(G) =

y ∂G

∂y , ∂G

∂x

1

, . . . , ∂G

∂x

n

O

_n+1+r

, J

_Σ_Λ_(G)

= ∂G

∂y , ∂G

∂x

1

, . . . , ∂G

∂x

n

, y

O

_n+1+r

.

As an example we take the simplest nontrivial case of type F

4

(cf. [7]):

g(y, x) = y

²

+ x

³

. Then G(y, x, u) = y

²

+ x

³

+ u

1

xy + u

2

y + u

3

x and

∆ e

²_Λ

(G) = h2y

²

+ yu

2

+ u

1

xy, 3x

²

+ u

1

y + u

3

i, J

ΣΛ(G)

= hu

2

+ u

1

x, 3x

²

+ u

3

, yi,

and also the quotient space

O

₂₊₃

∆ e

²_Λ

(G) + hyiJ

_Σ_Λ_(G)

is generated by {1, x, y, x

²

, x

³

, xy} as an O

3

-module.

We see that the functions

h(y, x, u) = α

1

(u) + α

2

(u)x + α

3

(u)x

²

+ α

4

(u)x

³

+ α

5

(u)y + α

6

xy + ψ(y, x, u)

(9)

with

α

5

(u) + α

6

(u)x ∈ J

_Σ_Λ_(G)

,

α

2

(u) + 2α

3

(u)x + 3α

4

(u)x

²

∈ J

_Σ_Λ_(G)

, ψ ∈ e ∆

²_Λ

(G) + hyiJ

_Σ_Λ_(G)

form the space H

G

.

Now it is easy to calculate the basis of vector fields tangent to Q

Λ

(G) (cf. [7]):

V

1

= −u

²₁

∂

∂u

2

+ 6u

2

∂

∂u

3

, V

2

= u

1

∂

∂u

1

+ u

2

∂

∂u

2

, V

3

= −u

1

∂

∂u

1

+ 2u

3

∂

∂u

3

, V

4

= 3u

2

∂

∂u

1

− u

₁

u

3

∂

∂u

2

, which satisfy the relation −u

1

V

4

+ u

3

V

1

− 3u

₂

V

3

= 0.

4.2. In the case of Λ singular our algorithm leads to quite complicated calculations. We show only some steps of the procedure which make clear the differences with the nonsingular case.

Let Λ = {(x, y) ∈ C

²

: F (x, y) = x

³

− y

²

= 0}. The module Ξ

Λ

of vector fields tangent to Λ is generated by

ξ

1

= 2x ∂

∂x + 3y ∂

∂y , ξ

2

= 3x

²

∂

∂y + 2y ∂

∂x .

We consider the simplest non-Morse function g(x, y) = x

³

+ y

²

. Its Jacobi ideal is

∆

Λ

(g) = hx

²

y, x

³

+ y

²

i and a versal deformation is

G(x, y, u) = x

³

+ y

²

+ u

1

xy

²

+ u

2

xy + u

3

x

²

+ u

4

y

²

+ u

5

x + u

6

y.

The corresponding Λ-bifurcation variety Q

Λ

(G) is described by the equations

3x

²

+ u

1

y

²

+ u

2

y + 2u

3

x + u

5

= 0, 2y + 2u

1

xy + u

2

x + 2u

4

y + u

6

= 0 together with x

³

− y

²

= 0.

The quotient space

O

2+6

∆ e

Λ

(G) + hx

³

− y

²

iJ

_Σ_Λ_(G)

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is generated by {1, x, y, x

²

, xy, y

²

, x

³

, x

²

y, xy

²

, y

³

, x

²

y

²

, xy

³

, y

⁴

} as an O

₆

- module. The functions

h(x, y, u) = α

0

(u) + α

1

(u)x + α

2

(u)y + α

3

(u)x

²

+ α

4

(u)xy + α

5

(u)y

²

+ α

6

(u)x

³

+ α

7

(u)x

²

y + α

8

(u)xy

²

+ α

9

(u)y

³

+ α

10

(u)x

²

y

²

+ α

11

(u)xy

³

+ α

12

(u)y

⁴

+ ψ(x, y, u)

with

α

1

(u) + 2α

3

(u)x + α

4

(u)y + 3α

6

(u)x

²

+ 2α

7

(u)xy

+ α

8

(u)y

²

+ 2α

10

(u)xy

²

+ +α

11

(u)y

³

∈ J

_Σ_Λ_(G)

, α

2

(u) + α

4

(u)x + 2α

5

(u)y + α

7

(u)x

²

+ 2α

8

(u)xy

+ 3α

9

(u)y

²

+ 2α

10

(u)x

²

y + 3α

11

(u)xy

²

+ 4α

12

(u)y

³

∈ J

_Σ_Λ_(G)

, ψ ∈ e ∆

²_Λ

(G) + hx

³

− y

²

iJ

_Σ_Λ_(G)

form the space H

G

.

Remarks. 1. If g is a Morse singularity on Λ singular then the Λ-bifurcation variety Q

Λ

(G) is diffeomorphic to the product Λ × C

^k

for some k ∈ N ∪ {0}.

2. Let G(x, u) be a germ of a holomorphic family of functions. Let Λ

0

⊂ C

ⁿ

be a germ of a complex space. We consider a deformation of Λ

0

, i.e. a family of varieties π : e Λ → C

^r

with π

⁻¹

(0) = Λ

0

. As a natural generalization of a Λ-bifurcation variety of G we have the e Λ-bifurcation variety of G defined by

Q

Λe

(G) =

(x, u) ∈ C

ⁿ

×C

^r

: ∂G

∂x

i

(x, u) = 0, (x, u) ∈ π

⁻¹

(u), i = 1, . . . , n

. If e Λ is the versal deformation of Λ

0

(cf. [8, 5]) we may use the normal forms of e Λ to consider the parametrized groups (deformations of groups) u → G

Λeu

, e Λ

u

= π

⁻¹

(u) acting on families G. In case of families of hypersurfaces, Λ is given by the holomorphic function F : (C e

ⁿ

× C

^r

, 0) → (C, 0), Λ

u

= {x ∈ C

ⁿ

: F (·, u) = 0}. So the classification problem of e Λ-varieties is reduced to the classification of map-germs (F, G) : (C

ⁿ

× C

^r

, 0) → C

²

with right and modified left equivalences (cf. [13]).

References

[1] V. I. A r n o l d, S. M. G u s e i n - Z a d e and A. N. V a r c h e n k o, Singularities of Dif- ferentiable Maps, Vol. 1, Birkh¨ auser, Boston, 1985.

[2] J. W. B r u c e, Functions on discriminants, J. London Math. Soc. (2) 30 (1984),

551–567.

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[3] J. W. B r u c e and R. M. R o b e r t s, Critical points of functions on analytic varieties, Topology 27 (1988), 57–90.

[4] V. G u i l l e m i n and S. S t e r n b e r g, Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge, 1984.

[5] S. I z u m i y a, Generic bifurcations of varieties, Manuscripta Math. 46 (1984), 137–164.

[6] S. J a n e c z k o, On isotropic submanifolds and evolution of quasicaustics, Pacific J.

of Math. 158 (1993), 317–334.

[7] —, On quasicaustics and their logarithmic vector fields, Bull. Austral. Math. Soc.

43 (1991), 365–376.

[8] A. K a s and M. S c h l e s s i n g e r, On the versal deformation of a complex space with an isolated singularity , Math. Ann. 196 (1972), 23–29.

[9] S. L o j a s i e w i c z, Introduction to Complex Analytic Geometry , Birkh¨ auser, 1991.

[10] O. W. L y a s h k o, Classification of critical points of functions on a manifold with singular boundary , Funktsional. Anal. i Prilozhen. 17 (3) (1983), 28–36 (in Russian).

[11] K. S a i t o, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265–291.

[12] H. T e r a o, The bifurcation set and logarithmic vector fields, Math. Ann. 263 (1983), 313–321.

[13] C. T. W a l l, A splitting theorem for maps into R

²

, ibid. 259 (1982), 443–453.

[14] V. M. Z a k a l y u k i n, Bifurcations of wavefronts depending on one parameter , Func- tional Anal. Appl. 10 (1976), 139–140.

INSTITUTE OF MATHEMATICS

WARSAW UNIVERSITY OF TECHNOLOGY PL. POLITECHNIKI 1

00-661 WARSZAWA, POLAND

Re¸ cu par la R´ edaction le 29.5.1995