The purpose of the paper is to illustrate how vanishing theorems can be used to give effective criteria for a generically finite morphism $f\,{:}\,X\,{\longrightarrow}\,Y$ 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 $P_2(X)\,{=}\,P_2(Y)$.