## Journal of the Mathematical Society of Japan

### Degeneracy locus of critical points of the distance function on a holomorphic foliation

#### Abstract

We study the geometry of transversality of holomorphic foliations of codimension one in ${\mathbb C}^n$ with spheres, from a viewpoint of dynamics of anti-holomorphic maps in the projective space. A point of non-degenerate contact of a leaf with a sphere is a hyperbolic fixed point of the corresponding dynamics. Around a point of degenerate contact, the intersection of branches of the variety of contacts is described as a bifurcation diagram of a neutral fixed point of dynamics. The Morse index for the distance function from the origin is computed as the complex dimension of an unstable manifold.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 1 (2014), 123-137.

Dates
First available in Project Euclid: 24 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1390600839

Digital Object Identifier
doi:10.2969/jmsj/06610123

Mathematical Reviews number (MathSciNet)
MR3161395

Zentralblatt MATH identifier
1288.32044

#### Citation

ITO, Toshikazu; SCÁRDUA, Bruno; YAMAGISHI, Yoshikazu. Degeneracy locus of critical points of the distance function on a holomorphic foliation. J. Math. Soc. Japan 66 (2014), no. 1, 123--137. doi:10.2969/jmsj/06610123. https://projecteuclid.org/euclid.jmsj/1390600839

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