Semistability of logarithmic cotangent bundle on some projective manifolds

Abstract

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 Ω_X (log D) 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 n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.