no. 2
On deformations of holomorphic foliations
Annales de l'Institut Fourier, Tome 39 (1989) no. 2, pp. 417-449.

Pour un feuilletage holomorphe non singulier ℱ sur une variété compacte M, nous comparons les espaces versels K et K tr des déformations de ℱ en feuilletages holomorphes et en feuilletages transversalement holomorphes, respectivement. Pour cela, nous prouvons l’existence d’un déploiement versel de ℱ paramétré par un espace analytique K f isomorphe à π -1 (0)×Σ, où Σ est lisse et π:K→K f est le morphisme d’oubli. On montre que l’application π est un épimorphisme dans deux situations : (i) si H 2 (M,Θ ℱ f )=0, où Θ ℱ f est le faisceau des germes de champs vectoriels holomorphes et tangents à ℱ, et (ii) s’il existe un feuilletage holomorphe ℱ ⋔ transverse et supplémentaire de ℱ. Quand les conditions (i) et (ii) sont toutes deux vérifiées, on a K≅K f ×K tr .

Given a non-singular holomorphic foliation ℱ on a compact manifold M we analyze the relationship between the versal spaces K and K tr of deformations of ℱ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of ℱ parametrized by an analytic space K f isomorphic to π -1 (0)×Σ where Σ is smooth and π : K→K tr is the forgetful map. The map π is shown to be an epimorphism in two situations: (i) if H 2 (M,Θ ℱ f )=0, where Θ ℱ f is the sheaf of germs of holomorphic vector fields tangent to ℱ, and (ii) if there exists a holomorphic foliation ℱ ⋔ transverse and supplementary to ℱ. When the conditions (i) and (ii) are both fulfilled then K≅K f ×K tr .

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     title = {On deformations of holomorphic foliations},
     journal = {Annales de l'Institut Fourier},
     pages = {417--449},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {2},
     year = {1989},
     doi = {10.5802/aif.1172},
     zbl = {0659.32019},
     mrnumber = {91b:32021},
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Girbau, Joan; Nicolau, Marcel. On deformations of holomorphic foliations. Annales de l'Institut Fourier, Tome 39 (1989) no. 2, pp. 417-449. doi : 10.5802/aif.1172. https://aif.centre-mersenne.org/articles/10.5802/aif.1172/

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