The LMDz variable grid GCM was used to simulate the Last Glacial Maximum (LGM, 21 ky Bp.) climate of Greenland and Antarctica at a spatial resolution of about 100 km.The high spatial resolution allows to investigate the spatial variability of surface climate change signals, and thus to address the question whether the sparse ice core data can be viewed as representative for the regional scale climate change. This study addresses primarily surface climate parameters because these can be checked against the, limited, ice core record. The changes are generally stronger for Greenland than for Antarctica, as the imposed changes of the forcing boundary conditions (e.g., sea surface temperatures) are more important in the vicinity of Greenland. Over Greenland, and to a limited extent also in Antarctica, the climate shows stronger changes in winter than in summer. The model suggests that the linear relationship between the surface temperature and inversion strength is modified during the LGM. The temperature dependency of the moisture holding capacity of the atmosphere alone cannot explain the strong reduction in snowfall over central Greenland; atmospheric circulation changes also play a crucial role. Changes in the high frequency variability of snowfall, atmospheric pressure and temperature are investigated and possible consequences for the interpretation of ice core records are discussed. Using an objective cyclone tracking scheme, the importance of changes of the atmospheric dynamics off the coasts of the ice sheets, especially for the high frequency variability of surface climate parameters, is illustrated. The importance of the choice of the LGM ice sheet topography is illustrated for Greenland, where two different topographies have been used, yielding results that differ quite strongly in certain nontrivial respects. This means that the paleo-topography is a significant source of uncertainty for the modelled paleoclimate. The sensitivity of the Greenland LGM climate to the prescribed sea surface conditions is examined by using two different LGM North Atlantic data sets.