Morse Theory and the topology of holomorphic foliations near an isolated singularity

TL;DR

This paper explores the topology of holomorphic foliations near isolated singularities in complex spaces using Morse theory and polar varieties, revealing new geometric insights into their structure.

Contribution

It introduces a novel approach combining Morse theory with polar varieties to analyze the local topology of holomorphic foliations at singularities.

Findings

Characterization of polar varieties associated with holomorphic foliations.

Insights into the change of leaf intersections with level sets of Morse functions.

Extension of classical algebraic geometry concepts to complex foliation topology.

Abstract

Let $\mathcal{F}$ be the germ at $\mathbf{0} \in \mathbb{C}^n$ of a holomorphic foliation of dimension $d$, $1 \leq d < n$, with an isolated singularity at $\mathbf{0}$. We study its geometry and topology using ideas that originate in the work of Thom concerning Morse theory for foliated manifolds. Given $\mathcal{F}$ and a real analytic function $g$ on $\mathbb{C}^n$ with a Morse critical point of index 0 at $\mathbf{0}$, we look at the corresponding polar variety $M= M(\mathcal{F},g)$. These are the points of contact of the two foliations, where $\mathcal{F}$ is tangent to the fibres of $g$. This is analogous to the usual theory of polar varieties in algebraic geometry, where holomorphic functions are studied by looking at the intersection of their fibers with those of a linear form. Here we replace the linear form by a real quadratic map, the Morse function $g$. We then study $\mathcal{F}$ by looking at the intersection of its leaves with the level sets of $g$, and the way how these intersections change as the sphere gets smaller.