Given a non-singular holomorphic foliation on a compact manifold we analyze the relationship between the versal spaces and of deformations of as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of parametrized by an analytic space isomorphic to where is smooth and : is the forgetful map. The map is shown to be an epimorphism in two situations: (i) if , where is the sheaf of germs of holomorphic vector fields tangent to , and (ii) if there exists a holomorphic foliation transverse and supplementary to . When the conditions (i) and (ii) are both fulfilled then .
Pour un feuilletage holomorphe non singulier sur une variété compacte , nous comparons les espaces versels et des déformations de en feuilletages holomorphes et en feuilletages transversalement holomorphes, respectivement. Pour cela, nous prouvons l’existence d’un déploiement versel de paramétré par un espace analytique isomorphe à , où est lisse et est le morphisme d’oubli. On montre que l’application est un épimorphisme dans deux situations : (i) si , où est le faisceau des germes de champs vectoriels holomorphes et tangents à , et (ii) s’il existe un feuilletage holomorphe transverse et supplémentaire de . Quand les conditions (i) et (ii) sont toutes deux vérifiées, on a .
@article{AIF_1989__39_2_417_0, author = {Girbau, Joan and Nicolau, Marcel}, 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}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1172/} }
TY - JOUR AU - Girbau, Joan AU - Nicolau, Marcel TI - On deformations of holomorphic foliations JO - Annales de l'Institut Fourier PY - 1989 SP - 417 EP - 449 VL - 39 IS - 2 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1172/ DO - 10.5802/aif.1172 LA - en ID - AIF_1989__39_2_417_0 ER -
%0 Journal Article %A Girbau, Joan %A Nicolau, Marcel %T On deformations of holomorphic foliations %J Annales de l'Institut Fourier %D 1989 %P 417-449 %V 39 %N 2 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.1172/ %R 10.5802/aif.1172 %G en %F AIF_1989__39_2_417_0
Girbau, Joan; Nicolau, Marcel. On deformations of holomorphic foliations. Annales de l'Institut Fourier, Volume 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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