Abstract:
This thesis is divided into two parts.
In the first part, it is proved that, 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. The present study investigates 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. Semistability of this sheaf, is proved, when the log canonical sheaf KX + 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 < or = 6. The study also extends the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.
The second part, investigates linear systems on hyper elliptic varieties. Analogues of well-known theorems on abelian varieties, like Lefschetz's embedding theorem and higher k-jet embedding theorems are proved. Syzygy or Np-properties are also deduced for appropriate powers of ample line bundles.
