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Vanishing holonomy and monodromy of certain centres and foci. (English) Zbl 0841.32020
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 37-48 (1994).
We consider a germ \(\omega = A(x,y) dx + B(x,y) dy\) of a real-analytic 1-form on \((\mathbb{R}^2, 0)\) with a monodromic semidegenerate singularity. The singularity of \(\omega\) is called monodromic semidegenerate if the first nonzero quasi-homogeneous jet of type \((1,2)\) of \(\omega\) is \(\omega_0 = x^3 dx + (y + ax^2) dy\) for some real number \(a\) with \(a^2 < 2\). Let \(P : (\mathbb{C}, 0) \to (\mathbb{C}, 0)\) denote the monodromy map of \(\omega\) corresponding to the embedding \((\mathbb{C}, 0) \to (\mathbb{C}^2, 0)\), \(t \mapsto (t,0)\). Let \(h : M \to \mathbb{C}^2\) be the composition of two blowing-ups. The center of this first blowing-up is the origin of \(\mathbb{C}^2\), and the center of the second is a real point on the exceptional curve of the first. It is easy to see that the pullback \(h^* \omega\) has two conjugate singularities of hyperbolic type on the exceptional curve of the second blowing-up, if we choose the center of the second blowing-up. Associated with these hyperbolic singularities we can define two holonomy maps \(f,g : (\mathbb{C}, 0) \to (\mathbb{C}, 0)\). It is known that \((fg)^2\) is the identity map. The following results have been obtained in this article: Theorem. \(P = [f,g]\). Corollary 1. Assume \(a \neq 0\) and that \(P\) is the identity map. Then, the germ \(\omega\) is analytically equivalent to \(\omega_0\). Corollary 2. If \(P\) is the identity map, then there exists a nontrivial analytic involution \(I : (\mathbb{R}^2, 0) \mapsto (\mathbb{R}^2, 0)\) with \(I^* \omega \wedge \omega = 0\).
For the entire collection see [Zbl 0797.00019].
Reviewer: T.Urabe (Tokyo)

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
57R30 Foliations in differential topology; geometric theory