  The supergravity (SUGRA) theories with exact global $U(1)$ symmetry or shift
symmetry in K\"ahler potential provide the natural frameworks for inflation.
However, the quadratic inflation is disfavoured by the new results on
primordial tensor fluctuations from the Planck Collaboration. To be consistent
with the new Planck data, we point out that the explicit symmetry breaking is
needed, and study these two SUGRA inflation in details. For the SUGRA inflation
with global $U(1)$ symmetry, the symmetry breaking term leads to a
trigonometric modulation on inflaton potential. The coefficient of the $U(1)$
symmetry breaking term is of the order $10^{-2}$, which is sufficient large to
improve the inflationary predictions while its higher order corrections are
negligible. Such models predict sizeable tensor fluctuations and highly agree
with the Planck results. In particular, the model with a linear $U(1)$ symmetry
breaking term predicts the tensor-to-scalar ratio around $\textbf{r}\sim0.01$
and running spectral index $\alpha_s\sim-0.004$, which comfortably fit with the
Planck observations. For the SUGRA inflation with breaking shift symmetry, the
inflaton potential is modulated by an exponential factor. The modulated linear
and quadratic models are consistent with the Planck observations. In both kinds
of models the tensor-to-scalar ratio can be of the order $10^{-2}$, which will
be tested by the near future observations.
