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Small field inflation in ${\cal N}=1$ supergravity with a single chiral superfield

We consider "new inflation" inflationary models at small fields, embedded in minimal ${\cal N}=1$ supergravity with a single chiral superfield. Imposing a period of inflation compatible with experiment severely restricts possible models, classified in perturbation theory. If moreover we impose that the field goes to large values and very small potential at the current time, like would be needed for instance for the inflaton being the volume modulus in large extra dimensional scenarios, the possible models are restricted to very contrived superpotentials.