  In the present paper we discuss stability of the tanget bundle of a Fano
n-fold of index >= n-2 and b_2=1. For example, we prove that all Fano 4-folds
with b_2=1 have stable tangent bundle. For this purpose we prove some vanishing
theorems for sheaves of twisted holomorphic forms on ample divisors and cyclic
coverings of manifolds having "special cohomology".
