Journal of the Mathematical Society of Japan

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

Toshikazu ITO, Bruno SCÁRDUA, and Yoshikazu YAMAGISHI

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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

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

First available in Project Euclid: 24 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32S65: Singularities of holomorphic vector fields and foliations
Secondary: 37F75: Holomorphic foliations and vector fields [See also 32M25, 32S65, 34Mxx]

foliation singularity transversality Morse index anti-holomorphic


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.

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