One of the fundamental problems of algebraic geometry is to classify complex projective manifolds up to birational isomorphism. When possible this classification is usually achieved in terms of the numbers P_m(X) (known as the m-th plurigenera of X) which correspond the number of linearly independent (m-multiple-valued) top-degree holomorphic differential forms on the manifold X. If P_m(X) is of maximal growth, one says that X is of general type. A natural problem that arises in this context is to determine explicit conditions for a surjective morphism f:X--->Y of n-dimensional manifolds to be a birational isomorphism. We will discuss the connection between this problem and a famous conjecture of Fujita. When $Y$ is of general type and has big fundamental group, there is a surprisingly simple answer: Theorem: Let f:X--->Y be a surjective morphism of n-dimensional manifolds. If $Y$ is of general type and of generically large algebraic fundamental group, then $f$ is a birational isomorphism if and only if $P_2(X)=P_2(Y)$.