Strong Topological Invariance of the Monodromy Group at Infinity for Quadratic Vector Fields

Journal of Singularities
volume 9 (2014), 193-202

Received 14 December 2011. Received in revised form 17 January 2013.

DOI: 10.5427/jsing.2014.9n

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Abstract:

In this work we consider foliations on CP^2 which are generated by quadratic vector fields on C^2. Generically these foliations have isolated singularities and an invariant line at infinity. We say that the monodromy groups at infinity of two such foliations having the same singular points at infinity are strongly analytically equivalent provided there exists a germ of a conformal mapping at zero which conjugates the monodromy maps defined along the same loops on the infinite leaf.

The object of this paper is to show that topologically equivalent generic foliations from this class must have, after an affine change of coordinates, their monodromy groups at infinity strongly analytically conjugated.

As a corollary it is proved that any two such generic and sufficiently close foliations can only be topologically conjugated if they are affine equivalent. This improves, in the case of quadratic vector fields, a result of Ilyashenko from 1978 which claims that two generic, topologically equivalent and sufficiently close foliations are affine equivalent provided the conjugating homeomorphism is close enough to the identity map.

Author(s) information:

Valente Ramírez