Signature of the Milnor fiber of parametrized surfaces

TL;DR

This paper derives an explicit algebraic formula for the signature of the Milnor fiber of certain non-isolated complex surface singularities, establishing it as a topological invariant and illustrating its application with numerous examples.

Contribution

It provides the first explicit algebraic formula for the Milnor fiber signature of finitely determined holomorphic germs, linking complex analytic and topological methods.

Findings

Signature is a topological invariant for these singularities

Explicit algebraic formula for the Milnor fiber signature

Numerous computed examples demonstrating the formula's use

Abstract

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Sz\H{u}cs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.