Singular Hermitian metrics and positivity of direct images of   pluricanonical bundles
  This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part. 
Mihai Paun
math.AG