Let be a complex-analytic -principal bundle where is a complex Lie group. The complex-analytic Ehresmann connection is the analytic field of horizontal subspaces, which is -equivariant. One can consider the underlying real principal bundle . The operator of complex structure becomes an automorphism of the smooth real -bundle . One can consider the differential
(d I)_p T_p E\to T_p E
which on each vertical subspace is an operator of the complex structure on the fiber.
For a field of horizontal subspaces on the following are equivalent:
(i) is a connection on
One can characterize complex connections also by conditions on a covariant derivative on .