  We determine the twistor deformation of rank one local systems on compact
Kaehler manifolds which correspond to smooth twistor modules of rank one in the
sense of C. Sabbah. Our proof is rather elementary, and uses a natural
description of the moduli space of rank one local systems together with the
canonical morphism to the Picard variety. The corresponding assertion for
smooth twistor modules of rank one follows from the theory of C. Simpson and
has been known to the specialists, according to T. Mochizuki.
