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On the density of the Pfaffian systems without algebraic solution. (Sur la densité des systèmes de Pfaff sans solution algébrique.) (French) Zbl 0792.58001
Let \(M\) be an analytic surface. A. Lins Neto [J. Differ. Geom. 26, 1-31 (1987; Zbl 0625.57012)] introduced a topology in the set \(\Pi(M)\) of holomorphic foliations with isolated singularities on \(M\).
\(\Omega \in \Pi(M)\) is “rigid” if it is an isolated point of \(\Pi(M)\). In our paper it is proved that if \(M\) is a projective rational surface non-isomorphic to \(\mathbb{P}_ 2(\mathbb{C})\) then there exists \(\Omega \in \Pi(M)\) rigid and having algebraic leaves.
The case of \(\mathbb{P}_ 2(\mathbb{C})\) has been considered by J. P. Jouanolou [‘Equations de Pfaff algébriques’ (1979; Zbl 0477.58002)].

MSC:
58A17 Pfaffian systems
57R30 Foliations in differential topology; geometric theory
32S65 Singularities of holomorphic vector fields and foliations
32S05 Local complex singularities
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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References:
[1] A. LINS NETO, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. · Zbl 0625.57012
[2] J.-P. JOUANOLOU, Equations de Pfaff algébriques, Lecture Notes in Math. Springer Verlag, vol. 708, 1979. · Zbl 0477.58002
[3] A. BEAUVILLE, Surfaces algébriques complexes, Astérisque, vol. 54 (1978). · Zbl 0394.14014
[4] C. CAMACHO et P. SAD, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math., 115 (1982), 579-595. · Zbl 0503.32007
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