  We analyse the implications of the Planck data for cosmic inflation. The
Planck nominal mission temperature anisotropy measurements, combined with the
WMAP large-angle polarization, constrain the scalar spectral index to $n_s =
0.9603 \pm 0.0073$, ruling out exact scale invariance at over 5 $\sigma$.
Planck establishes an upper bound on the tensor-to-scalar ratio of r < 0.11
(95% CL). The Planck data thus shrink the space of allowed standard
inflationary models, preferring potentials with V" < 0. Exponential potential
models, the simplest hybrid inflationary models, and monomial potential models
of degree n > 2 do not provide a good fit to the data. Planck does not find
statistically significant running of the scalar spectral index, obtaining $d
n_s/d ln k = -0.0134 \pm 0.0090$. Several analyses dropping the slow-roll
approximation are carried out, including detailed model comparison and
inflationary potential reconstruction. We also investigate whether the
primordial power spectrum contains any features. We find that models with a
parameterized oscillatory feature improve the fit $\chi^2$ by ~ 10; however,
Bayesian evidence does not prefer these models. We constrain several
single-field inflation models with generalized Lagrangians by combining power
spectrum data with bounds on $f_\mathrm{NL}$ measured by Planck. The fractional
primordial contribution of CDM isocurvature modes in the curvaton and axion
scenarios has upper bounds of 0.25% or 3.9% (95% CL), respectively. In models
with arbitrarily correlated CDM or neutrino isocurvature modes, an
anticorrelation can improve $\chi^2$ by approximatively 4 as a result of
slightly lowering the theoretical prediction for the $\ell<40$ multipoles
relative to the higher multipoles. Nonetheless, the data are consistent with
adiabatic initial conditions.
