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Non-Negative Deformations of Weighted Homogeneous Singularities NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES J. J. NUNO-BALLESTEROS,~ B. OREFICE-OKAMOTO,´ J. N. TOMAZELLA Abstract. We consider a weighted homogeneous germ of complex analytic variety (X; 0) ⊂ (Cn; 0) and a function germ f :(Cn; 0) ! (C; 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X; 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X; 0), and where (X; 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f :(X; 0) ! C. In the last part we give some formulas for the invariants in terms of the weights and the degrees of the polynomials. 1. Introduction The Milnor number is an important invariant in Singularity Theory. When it is defined, it reflects interesting geometric properties of the germ. For instance, if f :(Cn; 0) ! C is an isolated singularity function germ, then its generic fiber has the homotopy type of a bouquet of spheres of real dimension n − 1. The number of such spheres, denoted by µ(f), is the Milnor number of f (see [15]). L^e-Ramanujam ([12]) and Timourian ([23]) show that a family of hypersurfaces in (Cn; 0) (with n 6= 3) has constant Milnor number if and only if all the hypersurfaces in the family have the same topological type. In other words, the Milnor number is a topological invariant whose constancy in a family controls the topological triviality. Because of these facts it is important to know whether a family has constant Milnor number. In [24], Varchenko shows the following result. Theorem 1.1. If f :(Cn; 0) ! C is a weighted homogeneous polynomial with an isolated Pk singularity and if ft(x) = f(x) + i=1 σi(t)αi(x) is a deformation of f with σi(0) = 0, then the Milnor number of ft, µ(ft), is constant if and only if the weighted degree of αi is not smaller than the weighted degree of f, for all i = 1; : : : ; k. A deformation like the one in theorem 1.1 (the weighted degree of αi is not smaller than the weighted degree of f, for all i such that σi is not zero) is called a non-negative deformation. In this paper, we want to find necessary and sufficient conditions for a deformation to be non-negative in other cases where the Milnor number (or another similar invariant) is well defined. In the case of varieties with isolated singularity, the Milnor number is 2000 Mathematics Subject Classification. Primary 32S15; Secondary 58K60, 32S30. Key words and phrases. Milnor number, Bruce-Roberts number, non-negative deformations. The first author was partially supported by DGICYT Grant MTM2012{33073 and CAPES-PVE Grant 88881.062217/2014-01. The second author was partially supported by FAPESP Grants 2011/08877-3 and 2013/14014-3. The third author was partially supported by FAPESP Grant 2013/10856-0 and CNPq Grant 309626/2014-5. 1 2 J. J. NUNO-BALLESTEROS,~ B. OREFICE-OKAMOTO,´ J. N. TOMAZELLA known to be well defined for a space curve singularity [5], an isolated complete intersection singularity - ICIS [11], or an isolated determinantal singularity - IDS [19]. In the case of functions, on one hand we have the Bruce-Roberts number of a function f :(Cn; 0) ! C with respect to a variety (X; 0) ⊂ (Cn; 0) [3]. On the other hand, we can also consider the Milnor number of a function f :(X; 0) ! C, where (X; 0) is either a curve [9, 17] or an IDS [19]. Moreover, we also provide some formulas for these invariants in terms of the weights and the degrees of the polynomials. 2. Weighted homogeneous functions and varieties We use this section to present the basic definitions we will use in the work. Let us n denote by On the local ring of germs of complex analytic functions f :(C ; 0) ! C. We ∗ fix positive integer numbers w1; : : : ; wn 2 N such that gcd(w1; : : : ; wn) = 1. Assume that f 2 On is expressed as a convergent power series: X α f(x) = aαx α2Nn We say that f is weighted homogeneous if there exists d such that α1w1 + ··· + αnwn = d for all α such that aα 6= 0. In such a case, we say that d is the weighted degree of f and we write, wt(f) = d. Equivalently, f is a polynomial such that w1 wn d n f(t x1; : : : ; t xn) = t f(x1; : : : ; xn); 8x 2 C ; 8t 2 C: We say that a variety germ (X; 0) ⊂ (Cn; 0) is weighted homogeneous if it is the zero set of an ideal I ⊂ On which can be generated by weighted homogeneous germs φ1; : : : ; φp (with respect to the same weights, but with possibly different degrees). Now, this is equivalent to say that w1 wn (t x1; : : : ; t xn) 2 X; 8x 2 X; 8t 2 C: In particular, we take the convention that (Cn; 0) is itself weighted homogeneous. Asso- ciated with (X; 0) we can consider in a natural way the ring C[x1; : : : ; xn]= hφ1; : : : ; φpi as a graded ring with respect to the weights (w1; : : : ; wn) (see, for instance, [14, p. 93]). Definition 2.1. Let (X; 0) ⊂ (Cn; 0) be weighted homogeneous as above. We say that a polynomial germ f :(X; 0) ! C is weighted homogeneous if it is homogeneous in the graded ring C[x1; : : : ; xn]= hφ1; : : : ; φpi. In particular, there exists d 2 N such that w1 wn d f(t x1; : : : ; t xn) = t f(x1; : : : ; xn); 8x 2 X; 8t 2 C: In this case, we call d the weighted degree of f and write d = wt(f). Any non-zero polynomial germ f :(X; 0) ! C can be written in a unique way as: f = fd + fd+1 + ··· + fl; with fd 6= 0, where each fi is weighted homogeneous of degree i. We call fd the initial part of f and denote it by in(f). We say that f is semi-weighted homogeneous if in(f) has isolated singularity (that is, there exist a representative X of (X; 0) such that X n f0g is non-singular and in(f) is regular on X n f0g). NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES 3 In this paper, by a deformation of a map germ f :(X; 0) ! (Y; 0) we mean another map germ F :(C × X) ! (Y; 0) such that F (0; x) = f(x), for all x 2 X. We also assume that F is origin preserving, that is, F (t; 0) = 0 for all t, so we have a 1-parameter family of map germs ft :(X; 0) ! (Y; 0) given by ft(x) = F (t; x). By abuse of notation, sometimes it is usual to denote the deformation F just by ft. On the other hand, we can also consider ~ ~ the associated unfolding F :(C × X) ! (C × Y; 0) given by F (t; x) = (t; ft(x)). In the particular case of a polynomial function f :(X; 0) ! (C; 0), any polynomial deformation ft can be written as: k X (1) ft(x) = f(x) + σi(t)αi(x); i=1 for some αi :(X; 0) ! (C; 0) and σi :(C; 0) ! (C; 0) i = 1; : : : ; k. Definition 2.2. Let (X; 0) ⊂ (Cn; 0) be a weighted homogeneous variety germ. We say that a polynomial deformation ft of f :(X; 0) ! (C; 0) as in (1) is non-negative if wt(in(αi)) ≥ wt(in(f)), for i = 1; : : : ; k such that αi 6= 0. 3. The Bruce-Roberts number Let (X; 0) ⊂ (Cn; 0) be an analytic variety germ. Bruce and Roberts introduce in [3] a generalization of µ(f) which we call Bruce-Roberts number of f with respect to (X; 0). It is defined by On µBR(f; X) = dimC ; df(θX ) n where θX is the On-module of vector fields in (C ; 0) which are tangent to (X; 0) and df denotes the differential map of f. We remark that if X = ;, then µBR(f; X) = µ(f). Like the Milnor number of f, this number shows some geometric properties of f and X. n For instance, if we consider the group RX of biholomorphisms of (C ; 0) which preserve X, then f is finitely determined with respect to the action of RX on On if and only if µBR(f; X) is finite and, in this case, the codimension of the orbit of f under this action is equal to µBR(f; X) (see [3]). In [18], we show the following result, relating the Milnor and Bruce-Roberts numbers. Theorem 3.1. Let (X; 0) ⊂ (Cn; 0) be a weighted homogeneous hypersurface with an n isolated singularity and let f :(C ; 0) ! C be an RX -finitely determined function germ. Then, −1 µBR(f; X) = µ(f) + µ(X \ f (0); 0): The goal of this section is to provide a result like theorem 1.1 for the Bruce-Roberts number. In order to do this we will need the following related property. We recall that a n 0 deformation ft of a function germ f :(C ; 0) ! (C; 0) is called C -RX -trivial if there is a homeomorphism Ψ : (C × Cn) ! (C × Cn) which is an unfolding of the identity map in n C such that ft ◦ t = f and t preserves (X; 0) (i.e., t(X) = X for all t 2 C).
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