Torrential rains often occur in the western Mediterranean region during the fall season when the Mediterranean Sea is still warm. The Mediterranean Sea acts in moistening and warming the low level of the atmosphere. Then, the southerly to easterly flow that prevails before and during torrential rainfall events, transports the conditionally unstable air toward the coasts where the convection can develop often triggered by a flow interaction with orography. This study examines the sensitivity to the sea surface temperature (SST) of very short range (18--24 hours) high-resolution (2.4 km) forecasts of heavy precipitation events. Three torrential rainfall events were selected as representative of extreme rainfall events that occurred over southern France: two cases of quasi-stationary mesoscale convective systems and one other case characterized by a slow moving frontal perturbation. For each case, a number of runs is performed with the MESO-NH research model using several SST fields differing in their resolution or/and their average value over the Mediterranean basin. Results show that an increase (a decrease) of SST by several degrees, on average, intensifies (weakens) the convection. The convection can even be stopped with strong decrease of SST. Impacts on the stationary behavior of the systems have also been pointed out. A more detailed SST field influences the mesoscale patterns of the sea surface heat fluxes but have almost no significant effect on the convection and the low-level jets forecast. Eventually, the SST has been allowed to evolve during the runs through the action of the air-sea interface fluxes, resulting in local effects such as significant cooling of the SST beneath the low-level jet but almost no impact on the very short range forecasts of heavy precipitation.