On deformation with constant Milnor number and Newton polyhedron

Abstract

Abstract. We show that every µ-constant family of isolated hypersurface singularities satisfying a nondegeneracy condition in the sense of Kouchnirenko, is topologically trivial, also is equimultiple.Let f : (C n , 0) → (C, 0) be the germ of a holomorphic function with an isolated singularity. The Milnor number of a germ f , denoted by µ(f ), is algebraically defined as the dim O n /J(f ), where O n is the ring of complex analytic function germs : (C n , 0) → (C, 0) and J(f ) is the Jacobian ideal in O n generated by the partial derivatives {We recall that the multiplicity m(f ) is defined as the lowest degree in the power se-are germs of holomorphic functions. We use the notation F t (z) = F (z, t) when t is fixed. Let m t denote the multiplicity and µ t denote the Milnor number of F t at the origin. The deformation F is equimultiple (resp. µ-constant) if m 0 = m t (resp. µ 0 = µ t ) for small t. It is well-known by the result of Lê-Ramanujam [9]. that for n = 3, the topological type of the family F t is constant under µ-constant deformations. The question is still open for n = 3. However, under some additional assumption, positive answers have been given. For example, if F t is non-degenerate in the sense of Kouchnirenko [7] and the Newton boundary Γ(F t ) of F t is independent of t, i.e., Γ(F t ) = Γ(f ), it follows that µ * (F t ) is constant, and hence F t is topologically trivial (see [12,16] for details). Motivated by the Briançon-Speder µ-constant family F t (z) = z 5 1 + z 2 z 7 3 + z 15 2 + tz 1 z 6 3 , which is topologically trivial but not µ * -constant, M. Oka [13] shows that any non-degenerate family of type F (z, t) = f (z) + tz A for A = (A 1 , . . . , A n ) ∈ N n , where N is the set of nonnegative integers and z A = zn as usual, is topologically trivial, under the assumption of µ-constancy. Our purpose of this paper is to generalize this result, more precisely, we show that every µ-constant non-degenerate family F t with not necessarily Newton boundary Γ(F t ) independent of t, is topologically trivial. Moreover, we show that F is equimultiple, which gives a positive answer to a question of Zariski [19,5,14] for a nondegenerate family. To prove the main result (Theorem 1.1 below), we shall use the notion of (c)-regularity in stratification theory, introduced by K. Bekka in [3], which is weaker than Whitney regularity, nevertheless (c)-regularity implies topological triviality. First, we give a characterization of (c)-regularity (Theorem 2.1 below). By using it, we can show that the µ-constancy condition for a non-degenerate family implies Bekkas (c)-regularity condition and then obtain the topological triviality as a corollary.

The Łojasiewicz exponent in non-degenerate deformations of surface singularities

“…More precisely, we prove that in µ-constant non-degenerate families of surface singularities, the Łojasiewicz exponent is also constant. In a similar spirit, we remark that the constancy of the multiplicity (= the order of f t at 0) in µ-constant non-degenerate families of hypersurface singularities of any dimension, which is a particular case of the famous Zariski problem, has already been settled by Y. O. M. Abderrahmane [1].…”

On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities

“…The relation (1) ⇒ (3) follows. Indeed, by (1), the § For the definition of γ 2 f t ,z , which we have not encountered yet, we also refer the reader to [19].…”

“…Another well known class of isolated hypersurface singularities which satisfies the Zariski multiplicity conjecture is described by the following theorem due to Abderrahmane [1] and Saia and Tomazella [23]. Theorem 1.2 (Abderrahmane and Saia-Tomazella).…”

Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

“…. But in each case, (C1)-(C4), the Zariski multiplicity conjecture is true if we deal with families of isolated singularities (the references for that are [1], [6], [15], [17], [19], [21]). In other words, the family {h t + z N 1 1…”

“…form a μ-constant family of isolated singularities as t varies. Then, applying [7, Lemma 3.1], with the hyperplane z n = 0, gives that the family {h t + z N 1 1 The Lê numbers are intersection numbers of certain analytic cycles (socalled Lê cycles) with certain affine subspaces. The definition of the Lê cycles, in turn, is based on the notion of gap sheaf.…”