Removing instability of Polonyi-Starobinsky supergravity by adding FI term

Polonyi-Starobinsky (PS) supergravity is the N=1 supergravity model of Starobinsky inflation with spontaneous supersymmetry breaking (after inflation) due to Polonyi superfield, and inflaton belonging to a massive vector supermultiplet. The PS model is used for an explicit realization of the (super-heavy) gravitino dark matter scenario in cosmology. We find a potential instability in this model, and offer a mechanism for its removal by adding a Fayet-Iliopoulos (FI) term.