A finiteness criterion for compact varieties of surjective holomorphic mappings

It is well known that there exist at most finitely many surjective meromorphic mappings from any compact variety X onto a Riemann surface Y of genus ^2.There are several possibilities of generalizing this fact to higher dimensions.For instance, the same assertion holds, if Y is a variety of general type ), or if Y is Caratheodory-hyperbolic (Urata [12]).In [8], S. Lang raised the question whether the statement also carries over to algebraic varieties that are hyperbolic in the sense of Kobayashi.A partial answer to this problem has been given by J. Noguchi [11] who proved the finiteness assertion for hyperbolic Kahler manifolds Y with semi-positive canonical bundle.Employing a different approach, we shall show that the semi-positivity condition for K γ may be dropped however, the smoothness of Y as well as the Kahler condition still remain essential in our considerations.