Abstract
The minimal Starobinsky supergravity with inflaton (scalaron) and goldstino in a massive vector supermultiplet is coupled to the dilaton-axion chiral superfield with the no-scale K\"ahler potential and a superpotential. The Kachru-Kallosh-Linde-Trivedi-type superpotential with a constant term is used to stabilize dilaton and axion during inflation, but it is shown to lead to an instability. The instability is cured by adding the alternative Fayet-Iliopoulos (FI) term that does not lead to the gauged $R$ symmetry. Other stabilization mechanisms, based on the Wess-Zumino-type superpotential, are also studied in the presence of the FI term. A possible connection to the D3-brane models is briefly discussed too.