On deformations of holomorphic foliations

Girbau, Joan, and Nicolau, Marcel. "On deformations of holomorphic foliations." Annales de l'institut Fourier 39.2 (1989): 417-449. <http://eudml.org/doc/74836>.

@article{Girbau1989,
abstract = {Given a non-singular holomorphic foliation $\{\cal F\}$ on a compact manifold $M$ we analyze the relationship between the versal spaces $K$ and $K^\{\rm tr\}$ of deformations of $\{\cal F\}$ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of $\{\cal F\}$ parametrized by an analytic space $K^f$ isomorphic to $\pi ^\{-1\}(0)\times \Sigma $ where $\Sigma $ is smooth and $\pi $ : $K\rightarrow K^\{\rm tr\}$ is the forgetful map. The map $\pi $ is shown to be an epimorphism in two situations: (i) if $H^2(M,\Theta ^ f_\{\{\cal F\}\})=0$, where $\Theta ^ f_\{\{\cal F\}\}$ is the sheaf of germs of holomorphic vector fields tangent to $\{\cal F\}$, and (ii) if there exists a holomorphic foliation $\{\cal F\}^\pitchfork $ transverse and supplementary to $\{\cal F\}$. When the conditions (i) and (ii) are both fulfilled then $K\cong K^f\times K^\{\rm tr\}$.},
author = {Girbau, Joan, Nicolau, Marcel},
journal = {Annales de l'institut Fourier},
keywords = {holomorphic foliation; deformations; transversely holomorphic foliation},
language = {eng},
number = {2},
pages = {417-449},
publisher = {Association des Annales de l'Institut Fourier},
title = {On deformations of holomorphic foliations},
url = {http://eudml.org/doc/74836},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Girbau, Joan
AU - Nicolau, Marcel
TI - On deformations of holomorphic foliations
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 2
SP - 417
EP - 449
AB - Given a non-singular holomorphic foliation ${\cal F}$ on a compact manifold $M$ we analyze the relationship between the versal spaces $K$ and $K^{\rm tr}$ of deformations of ${\cal F}$ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of ${\cal F}$ parametrized by an analytic space $K^f$ isomorphic to $\pi ^{-1}(0)\times \Sigma $ where $\Sigma $ is smooth and $\pi $ : $K\rightarrow K^{\rm tr}$ is the forgetful map. The map $\pi $ is shown to be an epimorphism in two situations: (i) if $H^2(M,\Theta ^ f_{{\cal F}})=0$, where $\Theta ^ f_{{\cal F}}$ is the sheaf of germs of holomorphic vector fields tangent to ${\cal F}$, and (ii) if there exists a holomorphic foliation ${\cal F}^\pitchfork $ transverse and supplementary to ${\cal F}$. When the conditions (i) and (ii) are both fulfilled then $K\cong K^f\times K^{\rm tr}$.
LA - eng
KW - holomorphic foliation; deformations; transversely holomorphic foliation
UR - http://eudml.org/doc/74836
ER -