A large sample of calibration stars for Gaia: log g from Kepler and
  CoRoT
  Asteroseismic data can be used to determine surface gravities with precisions
of < 0.05 dex by using the global seismic quantities Deltanu and nu_max along
with Teff and [Fe/H]. Surface gravity is also one of the four stellar
properties to be derived by automatic analyses for 1 billion stars from Gaia
data (workpackage GSP_Phot). We explore seismic data from MS F, G, K stars
(solar-like stars) observed by Kepler as a potential calibration source for
methods that Gaia will use for object characterisation (log g). We calculate
log g for bright nearby stars for which radii and masses are known, and using
their global seismic quantities in a grid-based method, we determine an
asteroseismic log g to within 0.01 dex of the direct calculation, thus
validating the accuracy of our method. We find that errors in Teff and mainly
[Fe/H] can cause systematic errors of 0.02 dex. We then apply our method to a
list of 40 stars to deliver precise values of surface gravity, i.e. sigma <
0.02 dex, and we find agreement with recent literature values. Finally, we
explore the precision we expect in a sample of 400+ Kepler stars which have
their global seismic quantities measured. We find a mean uncertainty
(precision) on the order of <0.02 dex in log g over the full explored range 3.8
< log g < 4.6, with the mean value varying only with stellar magnitude (0.01 -
0.02 dex). We study sources of systematic errors in log g and find possible
biases on the order of 0.04 dex, independent of log g and magnitude, which
accounts for errors in the Teff and [Fe/H] measurements, as well as from using
a different grid-based method. We conclude that Kepler stars provide a wealth
of reliable information that can help to calibrate methods that Gaia will use,
in particular, for source characterisation with GSP_Phot where excellent
precision (small uncertainties) and accuracy in log g is obtained from seismic
data.
