Twistor deformation of rank one local systems

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.