Finiteness of multi-loop superstring amplitudes

Superstring amplitudes of an arbitrary genus are calculated through super-Schottky parameters by a summation over the fermion strings. For a calculation of divergent multi-loop fermion string amplitudes a supermodular invariant regularization procedure is used. A cancellation of divergences in the superstring amplitudes is established. Grassmann variables are integrated, the superstring amplitudes are obtained to be explicitly finite and modular invariant.