The geometric characteristics of the building have an important effect on the initiation of the upward leader. Based on a three-dimensional variable grid physical lightning model, in the case of buildings with different geometric characteristics and the peak current (I_P) of the lightning strike, the influence factors of upward positive leader initiation at the tip of buildings during the development of downward negative cloud-to-ground flash is discussed. In addition to finding that, the height of the building has a promoting effect on the initiation of the leader. The following conclusions have obtained. For the same I_P, the taller the building, the smaller the difference (η) of its geometric on the leader initiation; for the buildings of the same height, as I_P increases, the η becomes smaller. The stable leader at the tip of the cylindrical tower type is the easiest to initiate and the closest to the needle, the smallest η have reached at 12.2%. The stable leader at tip of the cuboid type is the latest to initiates, the largest η have reached at 53.4%. For the case of short building and weaker I_P, the geometric characteristics of tip (tower or slope) has a significant effect on the leader initiation.