  The observed amplitude of the rotational photometric modulation of a star
with spots should depend on the inclination of its rotational axis relative to
our line of sight. Therefore, the distribution of observed rotational
amplitudes of a large sample of stars depends on the distribution of their
projected axes of rotation. Thus, comparison of the stellar rotational
amplitudes of the Kepler KOIs with those of Kepler single stars can provide a
measure to indirectly infer the properties of the spin-orbit obliquity of
Kepler planets. We apply this technique to the large samples of 993 KOIs and
33,614 single Kepler stars in temperature range of 3500-6500 K. We find with
high significance that the amplitudes of cool KOIs are larger, on the order of
10%, than those of the single stars. In contrast, the amplitudes of hot KOIs
are systematically lower. After correcting for an observational bias, we
estimate that the amplitudes of the hot KOIs are smaller than the single stars
by about the same factor of 10%. The border line between the relatively larger
and smaller amplitudes, relative to the amplitudes of the single stars, occurs
at about 6000K. Our results suggest that the cool stars have their planets
aligned with their stellar rotation, while the planets around hot stars have
large obliquities, consistent with the findings of Winn et al. (2010) and
Albrecht et al. (2012). We show that the low obliquity of the planets around
cool stars extends up to at least 50 days, a feature that is not expected in
the framework of a model that assumes the low obliquity is due to planet-star
tidal realignment.
