Blow-analytic retraction onto the central fibre

TL;DR

This paper introduces a method to construct a retraction of a neighborhood onto the central fiber in complex analytic spaces using blowanalytic techniques, applicable also in real analytic settings.

Contribution

It adapts Kuo's blowanalytic methods to create a retraction via a real analytic vector field flow, linking it to specialization maps and Milnor fibrations.

Findings

Constructed a neighborhood retraction onto the central fiber.

Connected the retraction with specialization maps and Milnor fibrations.

Method applicable in both complex and real analytic categories.

Abstract

Let X be a complex analytic space and let f:X -> C be a proper complex analytic function with nonsingular generic fibres. By adapting the blowanalytic methods of Kuo we construct a retraction of a neighbourhood of the central fibre f^{-1}(0) onto f^{-1}(0). Our retraction is defined by the flow of a real analytic vector field on an oriented real analytic blow-up of X. Then we describe in terms of this blow-up the associated specialization map and local Milnor fibrations. The method also works in real analytic category.