I read in a few places that for a hermitian line bundle endowed with a hermitian connection, the local connection 1-forms are purely imaginary-valued. I tried to prove this, and came up with something contradictory. In fact, what I did is exactly what Pr. O'Farrill did in this post:

Connections with compatible Hermitian products on complex line bundles

His last equation $\beta + \overline{\beta}=dh$ tells me that the real part of the connection 1-form $\beta$ generally does not vanish.

On the other hand, I found that contrary statement in Demailly's Complex analytic and differential geometry book p.299, and in D. Auroux's online lecture notes. See Lecture 10 at

http://ocw.mit.edu/courses/mathematics/18-966-geometry-of-manifolds-spring-2007/lecture-notes/

Thanks for helping to clear out my confusion.