On the topology and differential geometry of Kahler threefolds

In the first part of my thesis we provide infinitely many examples of pairs of diffeomorphic, non simply connected Kahler manifolds of complex dimension 3 with different Kodaira dimensions. Also, in any allowed Kodaira dimension we find infinitely many pairs of non deformation equivalent, diffeomorphic Kahler threefolds. In the second part we study the existence of Kahler metrics of positive total scalar curvature on 3-folds of negative Kodaira dimension. We give a positive answer for rationally connected threefolds. The proof relies on the Mori theory of minimal models, the weak factorization theorem and on a specialization technique.