nLab 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 $\xi_{\mathbf{R}}$. The operator of complex structure becomes an automorphism $I:\xi_{\mathbf{R}}\to \xi_{\mathbf{R}}$ of the smooth real $G_{\mathbf{R}}$-bundle $\xi_{\mathbf{R}}$. One can consider the differential

$$(d I)_p T_p E\to T_p E$$

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

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

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

(ii) $I^* H = H$

(iii) $H_p = (d I)_p H_p$

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

• M M Postnikov, Lectures on geometry, vol. III, lec. 10