On injectivity, vanishing and torsion-free theorems for algebraic varieties

We give a short and almost self-contained proof of generalizations of Kollar's vanishing and torsion-free theorems. Although they are contained in Ambro's much more general results on embedded normal crossing pairs, we give an alternate and direct reduction argument, to the mixed Hodge theory. In this sense, this paper gives a more readable account of the application to the log minimal model program for log canonical pairs.