Invariants of topological relative right equivalences

Abstract

Let (V, 0) be the germ of an analytic variety in C n and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce-Roberts number, μ B R (V, f ). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce-Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce-Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.

Non-Negative Deformations of Weighted Homogeneous Singularities

“…In [1], it is shown that if (X, 0) is a weighted homogeneous hypersurface with an isolated singularity and if f and g are C 0 -R X -equivalent function germs then μ BR (f, X) = μ BR (g, X). The following example shows that the converse of this result is not true.…”

“…THEOREM 3.6. Let (X, 0) ⊂ ‫ރ(‬ n , 0) be a weighted homogeneous hypersurface with isolated singularity, let f : ‫ރ(‬ n , 0) → ‫ރ‬ be a weighted homogeneous R X -finitely determined function and let f t be a deformation of f as in (1). The following statements are equivalent:…”

“…Let (X, 0) ⊂ ‫ރ(‬ n , 0) be a weighted homogeneous curve and let f : (X, 0) → ‫ރ‬ be a finite map germ. Let f t be a deformation of f as in (1). Then, μ(f t ) is constant if and only if f t | X j is a non-negative deformation of f | X j , for each branch X j of X.…”

“…THEOREM 4.4. Let (X, 0) ⊂ ‫ރ(‬ n , 0) be a weighted homogeneous ICIS and let f : (X, 0) → ‫ރ‬ be a weighted homogeneous germ with an isolated singularity and let f t be a deformation of f as in (1). If f t is non-negative, then μ(f t ) is constant.…”

Mixed Bruce–Roberts numbers

“…. , μ (1) (f ), μ (0) (f )), where μ (i) (f ) denotes the Milnor number of the restriction of f to a generic linear subspace of dimension i of C n , for i = 1, . .…”