On Homotopy Types of Complements of Analytic Sets and Milnor Fibres

Blow-analytic Retraction onto the Central Fibre

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(0) onto f(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 b...

Hypersurface complements , Milnor fibers and higher homotopy groups of arrangments

Morse Theory, Milnor Fibers and Minimality of Hyperplane Arrangements

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of p-cells equals the p-th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when ...

Singular Cohomology of the Analytic Milnor Fiber, and Mixed Hodge Structure on the Nearby Cohomology

We describe the homotopy type of the analytic Milnor fiber in terms of the special fiber of a strictly semi-stable reduction, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. In the appendix, we consider the de Rham cohomology of the analytic Milnor fiber of an isolated singularity, and we re-interpret some class...

Morse theory, Milnor fibers and hyperplane arrangements

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW complex in which the number of p-cells equals the p-th betti number.