  We show how we can implement within Supergravity chaotic inflation in the
presence of a pole of order one or two in the kinetic mixing of the inflaton
sector. This pole arises due to the selected logarithmic Kahler potentials K,
which parameterize hyperbolic manifolds with scalar curvature related to the
coefficient -N of the logarithmic term. The associated superpotential W
exhibits the same R charge with the inflaton-accompanying superfield and
includes all the allowed terms. The role of the inflaton can be played by a
gauge singlet or non-singlet superfield. Models with one logarithmic term in K
for the inflaton, require N=2 and some tuning, of the order of 10^-5, between
the terms of W and predict a tensor-to-scalar ratio r at the level of 0.001.
The tuning can be totally eluded for more structured K's, with N values
increasing with r and spectral index close or even equal to its present central
observational value.
