  The gravitino problem gives a severe constraint on the thermal leptogenesis
scenario. This problem leads us to consider some alternatives to it if we try
to keep the gravitino mass around the weak scale $m_{3/2} \sim 100$ GeV. We
consider, in this paper, the non-thermal leptogenesis scenario in the framework
of a minimal supersymmetric SO(10) model. Even if we start with the same
minimal SO(10) model, we have different predictions for low-energy
phenomenologies dependent on the types of seesaw mechanism. This is the case
for leptogenesis: it is shown that the type-I see-saw model gives a consistent
scenario for the non-thermal leptogenesis but not for type-II. The predicted
inflaton mass needed to produce the observed baryon asymmetry of the universe
is found to be $M_I \sim 5 \times 10^{11}$ GeV for the reheating temperature
$T_R = 10^6$ GeV.
