Foliations with a Morse center

Journal of Singularities
volume 9 (2014), 82-100

Received 3 April 2012. Received in revised form 16 December 2013.

DOI: 10.5427/jsing.2014.9h

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Abstract:

We say that a holomorphic foliation F on a complex surface M has a Morse center at p in M if F has a local first integral with a Morse singularity at p. Given a line bundle L on M, let Fol(M,L) be the set of foliations F on M such that T^*(F)=L, and let Fol_C(M,L) be the closure of the set of F in Fol(M,L) such that F has a Morse center. In the first result of this paper we prove that Fol_C(M,L) is an algebraic subset of Fol(M,L). We apply this result to prove the persistence of more than one Morse center for some known examples, as for instance the logarithmic and pull-back foliations. As an application, we give a simple proof that R(1,d+1) is an irreducible component of the space of foliations of degree d with a Morse center on P^2, where R(m,n) denotes the space of foliations with a rational first integral of the form f^m/g^n with m dg(f)=n dg(g).

Keywords:

holomorphic foliation, Morse center

Mathematical Subject Classification:

37F75, 34M15

Author(s) information:

A. Lins Neto
Instituto de Matemática Pura e Aplicada
Estrada Dona Castorina, 110
Horto, Rio de Janeiro, Brasil
email: alcides@impa.br