  We present a first internal delensing of CMB maps, both in temperature and
polarization, using the public foreground-cleaned (SMICA) Planck 2015 maps.
After forming quadratic estimates of the lensing potential, we use the
corresponding displacement field to undo the lensing on the same data. We build
differences of the delensed spectra to the original data spectra specifically
to look for delensing signatures. After taking into account reconstruction
noise biases in the delensed spectra, we find an expected sharpening of the
power spectrum acoustic peaks with a delensing efficiency of $29\,\%$ ($TT$)
$25\,\%$ ($TE$) and $22\,\%$ ($EE$). The detection significance of the
delensing effects is very high in all spectra: $12\,\sigma$ in $EE$
polarization; $18\,\sigma$ in $TE$; and $20\,\sigma$ in $TT$. The null
hypothesis of no lensing in the maps is rejected at $26\,\sigma$. While direct
detection of the power in lensing $B$-modes themselves is not possible at high
significance at Planck noise levels, we do detect (at $4.5\,\sigma$ under the
null hypothesis) delensing effects in the $B$-mode map, with $7\,\%$ reduction
in lensing power. Our results provide a first demonstration of polarization
delensing, and generally of internal CMB delensing, and stand in agreement with
the baseline $\Lambda$CDM Planck 2015 cosmology expectations.
