On the connectivity of Milnor fiber for mixed functions

TL;DR

This paper proves the connectivity of Milnor fibers for mixed polynomials under certain conditions, providing a new proof for the connectivity of Milnor fibers in non-reduced complex analytic functions.

Contribution

It establishes the connectivity of Milnor fibers for mixed functions assuming a sequence of smooth points converging to zero, offering a new proof for non-reduced complex functions.

Findings

Milnor fiber connectivity for mixed polynomials proven.

New proof for connectivity of Milnor fibers in non-reduced complex functions.

Assumption of smooth points sequence is key.

Abstract

In this note, we prove the connectivity of the Milnor fiber for a mixed polynomial $f(\mathbf z,\bar{\mathbf z})$, assuming the existence of a sequence of smooth points of $f^{-1}(0)$ converging to the origin. This result gives also a another proof for the connectivity of the Milnor fiber of a non-reduced complex analytic function which is proved by A. Dimca