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 $(-N)<0$ of a 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 $N=2$, 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.