Produção Nacional
Revisado por pares
A structural theorem for codimension one foliations on $\p^n$, $n\ge3$, with an application to degree three foliations
2013; Linguagem: Inglês
10.2422/2036-2145.201010_009
ISSN2036-2145
AutoresDominique Cerveau, Alcides Lins Neto,
Tópico(s)Point processes and geometric inequalities
ResumoLet F be a codimension-one foliation on P n : for each point p ∈ P n we define J (F, p) as the order of the first non-zero jet j k p (ω) of a holomorphic 1form ω defining F at p.The singular set of F is sing(F) = { p ∈ P n | J (F, p) ≥ 1}.We prove (main Theorem 1.2) that a foliation F satisfying J (F, p) ≤ 1 for all p ∈ P n has a non-constant rational first integral.Using this fact we are able to prove that any foliation of degree-three on P n , with n ≥ 3, is either the pull-back of a foliation on P 2 , or has a transverse affine structure with poles.This extends previous results for foliations of degree at most two.Mathematics Subject Classification (2010): 37FF75 (primary); 34M45 (secondary). Notation1. O n : the ring of germs at 0 ∈ C n of holomorphic functions.[ f, g] 0 : the intersection number of f, g ∈ m 2 \ {0}, when f and g have no common factor.5. < f, g >: the ideal generated by f, g ∈ O p . 6. Diff(C n , p): the group of germs at p ∈ C n of biholomorphisms f with f ( p) = p. 7. i X (ω): the interior product of the vector field X and the form ω. 8. L X : the Lie derivativative in the direction of the vector field X. 9. j k p : the k th -jet at the point p.
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