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Let ℱ be the germ at  0 ∈ ℂ  n   of a holomorphic foliation of dimension  d , 1⩽ d < n , with an isolated singularity at  0 . We study its geometry and topology using ideas that originate in the work of Thom concerning Morse theory for foliated manifolds. Given ℱ and a real analytic function  g  on ℂ  n   with a Morse critical point of index 0 at  0 , we look at the corresponding polar variety  M = M (ℱ,  g ). These are the points of contact of the two foliations, where 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 fibres with those of a linear form. Here we replace the linear form by a real quadratic map, the Morse function  g . We then study ℱ by looking at the intersection of its leaves with the level sets of  g , and the way in which these intersections change as the sphere gets smaller.

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