Effective Criteria for Birational Morphisms

The purpose of the paper is to illustrate how vanishing theorems can be used to give effective criteria for a generically finite morphism f : X → Y of smooth complex projective algebraic varieties to be birational. In particular, as a consequence of a non‐vanishing theorem of Kollár, it is shown that if Y is of general type and has generically large algebraic fundamental group, then f is birational if and only if P 2 ( X )= P 2 ( Y ).