Automorphisms and non-integrability

Abstract

On this note we prove that a holomorphic foliation of the projective plane with rich, but finite, automorphism group does not have invariant algebraic curves.<br>Seja {mathcal F} uma folheação do plano projetivo complexo de grau d com grupo de automorfismo finito e cuja ação no espaço de cofatores não possui ponto fixo. Neste artigo mostramos que se {mathcal F} possui ao menos uma singularidade genérica então {mathcal F} não possui nenhuma curva algébrica invariante

Rational constants of monomial derivations

“…We prove also (Proposition 9.3) that if d : k(x, y, z) → k(x, y, z) is a derivation such that d(x) = y p , d(y) = z q , d(z) = x r , where p, q, r ∈ N, then k(x, y, z) d = k if and only if pqr 2. Some similar questions are studied in the interesting paper [18]. (The authors wish to thank the referee for pointing out this paper.…”

Closed meromorphic 1-forms

“…Theorem 4.16 admits many generalizations, see for instance [41], [70], [18], [51]. Even the original statement has now many different proofs exploiting different aspects of the theory of algebraic foliations like automorphism groups of foliations [69,53]), global geometry of the space of foliations ( [50]), or arithmetic through Galois group actions ( [17]) to name a few.…”

Foliations on the projective plane with finite group of symmetries

“…The presence of symmetries has been used to understand some relevant problems in holomorphic foliation theory, especially concerning integrability. For instance, it is used in the construction of Jouanolou's foliations [11], which play an important role in his proof of the density of foliations on the complex projective plane P 2 without invariant algebraic curves; in [16] this relation between automorphism groups and integrability was explored and put into more concrete terms.…”