  We consider the basic supersymmetric (SUSY) models of F-term hybrid inflation
(FHI). We show that a simple class of Kaehler potentials ensures a resolution
to the \eta problem and allows for inflation of hilltop type. As a consequence,
observationally acceptable values for the spectral index, ns, can be achieved
constraining the coefficient c4k of the quartic supergravity correction to the
inflationary potential. For about the central value of ns, in the case of
standard FHI, the grand unification (GUT) scale turns out to be well below its
SUSY value with the relevant coupling constant \kappa in the range
(0.0006-0.15) and c4k=-(1100-0.05). In the case of shifted [smooth] FHI, the
GUT scale can be identified with its SUSY value for c4k=-16 [c4k=-1/16].

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory