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**Universal families of foliations by curves.**
*(English)*
Zbl 0641.32014

Singularités d’équations différentielles, Journ. Dijon/France 1985, Astérisque 150/151, 109-129 (1987).

[For the entire collection see Zbl 0623.00011.]

This paper represents a continuation and extension of the author’s previous work on foliations of complex analytic spaces by curves [cf. the author, Aportaciones Mat., Notas Invest. 1, 36-64 (1985; Zbl 0627.14011)]. In the first section, he recapitulates the concept of a geometric, possibly singular foliation of a complex manifold by holomorphic curves, as well as its analytic properties [cf. reference above]. From this he derives a more general, purely analytic definition: A holomorphic foliation (with singularities) of a complex manifold M is a holomorphic map \(X: L\to TM\) from a line bundle \(L\in Pic(M)\) to the tangent bundle TM of M. The equivalence of two such holomorphic foliations, (L,X) and (L’,X’), is just given by biholomorphic equivalence.

In Section 2, the author proves that there is a one-to-one correspondence between the holomorphic foliations of M by curves and the invertible subsheaves of the sheaf of holomorphic vector fields on M. This allows to define analytically parametrized families of such foliations and, by applying A. Douady’s theorem on the existence of universal families of quotient sheaves, to prove the existence of a universal family of holomorphic foliations of M by curves.

Finally, Section 3 provides some insight into the geometry of the parameter spaces of the constructed universal families. It is shown that the Chern class of the line bundle tangent to a foliation remains constant on each connected component of the universal parameter space, furthermore that for a compact Kähler manifold M with vanishing first Betti number the universal parameter space is a disjoint union of projective spaces, and that for a projective manifold M the universal space of foliations with given Chern class is a projective variety. As for the latter case, it is shown, in addition, that for some fixed Chern classes the universal space of foliations admits the structure of a projective bundle over a complex torus, and its dimension is computed. The basic ingredients of proof are the Kodaira-Nakano vanishing theorem, the Hirzebruch-Riemann-Roch theorem, and the existence of the Poincaré bundle on Picard varieties.

This paper represents a continuation and extension of the author’s previous work on foliations of complex analytic spaces by curves [cf. the author, Aportaciones Mat., Notas Invest. 1, 36-64 (1985; Zbl 0627.14011)]. In the first section, he recapitulates the concept of a geometric, possibly singular foliation of a complex manifold by holomorphic curves, as well as its analytic properties [cf. reference above]. From this he derives a more general, purely analytic definition: A holomorphic foliation (with singularities) of a complex manifold M is a holomorphic map \(X: L\to TM\) from a line bundle \(L\in Pic(M)\) to the tangent bundle TM of M. The equivalence of two such holomorphic foliations, (L,X) and (L’,X’), is just given by biholomorphic equivalence.

In Section 2, the author proves that there is a one-to-one correspondence between the holomorphic foliations of M by curves and the invertible subsheaves of the sheaf of holomorphic vector fields on M. This allows to define analytically parametrized families of such foliations and, by applying A. Douady’s theorem on the existence of universal families of quotient sheaves, to prove the existence of a universal family of holomorphic foliations of M by curves.

Finally, Section 3 provides some insight into the geometry of the parameter spaces of the constructed universal families. It is shown that the Chern class of the line bundle tangent to a foliation remains constant on each connected component of the universal parameter space, furthermore that for a compact Kähler manifold M with vanishing first Betti number the universal parameter space is a disjoint union of projective spaces, and that for a projective manifold M the universal space of foliations with given Chern class is a projective variety. As for the latter case, it is shown, in addition, that for some fixed Chern classes the universal space of foliations admits the structure of a projective bundle over a complex torus, and its dimension is computed. The basic ingredients of proof are the Kodaira-Nakano vanishing theorem, the Hirzebruch-Riemann-Roch theorem, and the existence of the Poincaré bundle on Picard varieties.

Reviewer: W.Kleinert

### MSC:

32G10 | Deformations of submanifolds and subspaces |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

14C05 | Parametrization (Chow and Hilbert schemes) |

57R30 | Foliations in differential topology; geometric theory |