The equivariant signature of hypersurface singularities and eta-invariant

Abstract

LETS be an analytic germ of a hypersurface singularity. The associated monodromy action preserves the intersection form of its Milnor fibre. In this paper we study the corresponding equivariant signatures. The guiding principle is the following. Write f as a composed singularity f = p 0 4, where 4 (resp. p) is an isolated complete intersection (resp. curve) singularity; express the equivariant signature off as the equivariant signature of p with coefficients in a (non-degenerate) hermitian flat bundle; identify this with an index of the signature bundle; and when the primary invariant (Chern class) vanishes, express it in terms of the eta-invariant of the boundary (of the Milnor fiber of p) with coefficients in the corresponding signature bundle. In the realization of this program, we have two basic obstructions: The monodromy action of p (resp.f) is not compact (finite), and the monodromy representation of 4 (i.e. the candidate for the flat bundle), in general, is degenerate. This second obstruction is solved by introducing the variation map of 4. The variation structure (i.e. the degenerate monodromy representation together with the variation map) substitutes perfectly the non-degenerate representations. It turns out that the equivariant signature can be expressed in terms of this variation structure and the geometry of the curve singularity p. It has a sum decomposition, corresponding to the Jaco-Shalen-Johansson (or splice) decomposition of the link complement of p- ‘(0); each term is closely related to the Seifert geometry of the components. (The structure of the fundamental group of the Seifert components will remove the first obstruction too.) The primary invariant (here the first Chern class) vanishes when the variation structure is abelian or the intersection form of 4 is definite. The basic application for the first case is the topological series; the coverings exemplify the second case. The equivariant signature has been computed only for a few families of isolated singularities: curve singularities [ 151, quasi-homogeneous germs [20], suspensions [ 131. This paper gives the variable term of the composed topological series (in particular for the Yomdin’s series) (see Corollaries 5.4 and 5.11), and reduces the general case (see 5.1) to a signature computation of a non-degenerate hermitian flat bundle over the r-punctured 2-dimensional sphere (for which there exists a clear algorithm [S]).