Gravitino Problem in Inflation Driven by Inflaton-Polonyi K\"ahler Coupling

We discuss the cosmological gravitino problem in inflation models in which the inflaton potential is constructed from K"ahler potential rather than superpotential: a representative model is $\overline{\text{D}3}$-induced geometric inflation. A critical ingredient in this type of models is the coupling of the inflaton and Polonyi (supersymmetry-breaking) field in the K"ahler potential, which is needed to build the inflaton potential. We point out the same coupling let the inflaton dominantly decay into a pair of inflatino and gravitino causing the gravitino problem. We propose some possible solutions to this problem.