SEMISTABILITY OF LOGARITHMIC COTANGENT BUNDLE ON SOME PROJECTIVE MANIFOLDS

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kahler-Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf (X)(logD) on X. We prove semistability of this sheaf, when the log canonical sheaf K-X+D is ample or trivial, or when -K-X-D is ample, i.e., when X is a log Fano n-fold of dimension n6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.