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.
Categories: math.CV