  The minimal supersymmetric (or F-term) hybrid inflation is defined by a
unique renormalizable superpotential, fixed by a $U(1)$ R-symmetry, and it
employs a canonical K\"{a}hler potential. The inflationary potential takes into
account both radiative and supergravity corrections, as well as an important
soft supersymmetry breaking term, with a mass coefficient in the range $(0.1 -
10)~{\rm TeV}$. The latter term assists in obtaining a scalar spectral index
$n_s$ close to $0.96$, as strongly suggested by the PLANCK and WMAP-9yr
measurements. The minimal model predicts that the tensor-to-scalar $r$ is
extremely tiny, of order $10^{-12}$, while the spectral index running,
$|dn_{\rm s}/d\ln k| \sim 10^{-4}$. If inflation is associated with the
breaking of a local $U(1)_{B-L}$ symmetry, the corresponding symmetry breaking
scale $M$ is $(0.7 - 1.6)\cdot10^{15}~{\rm GeV}$ with $n_{\rm s} \simeq 0.96$.
This scenario is compatible with the bounds on $M$ from cosmic strings, formed
at the end of inflation from $B-L$ symmetry breaking. We briefly discuss
non-thermal leptogenesis which is readily implemented in this class of models.
