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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 $$\mathbb{C}$$
##### Keywords:
holomorphic foliations; projective rational surface
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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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