complex connection

Let $\xi =(p:E\to B)$ be a complex-analytic $G$-principal bundle where $G$ is a complex Lie group. The complex-analytic Ehresmann connection is the analytic field of horizontal subspaces, which is $G$-equivariant. One can consider the underlying real principal bundle ${\xi}_{R}$. The operator of complex structure becomes an automorphism $I:{\xi}_{R}\to {\xi}_{R}$ of the smooth real ${G}_{R}$-bundle ${\xi}_{R}$. One can consider the differential

$$(dI{)}_{p}{T}_{p}E\to {T}_{p}E$$

which on each vertical subspace $({T}_{p}^{V}E{)}_{R}$ is an operator of the complex structure on the fiber.

For a field $H$ of horizontal subspaces on ${\xi}_{R}$ the following are equivalent:

(i) $H$ is a connection on $\xi $

(ii) ${I}^{*}H=H$

(iii) ${H}_{p}=(dI{)}_{p}{H}_{p}$

One can characterize complex connections also by conditions on a covariant derivative on ${\xi}_{R}$.

- M M Postnikov,
*Lectures on geometry*, vol. III, lec. 10

Created on January 30, 2012 17:31:15
by Zoran Škoda
(161.53.130.104)