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We show how two topologically distinct spaces-the Kahler K3 x T^ 2 and the non-Kahler T^ 2 bundle over K3-can be smoothly connected in heterotic string theory. The transition occurs when the base K3 is deformed to the T^ 4/Z_2 orbifold limit. The orbifold theory can be mapped via duality to M-theory on K3 x K3 where the transition corresponds to an exchange of the two K3's.

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern classes to correspond to stable bundles.

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and HirzebruchRiemannRoch, and some modern development of uniformization theorems, KhlerEinstein metric and the theory of DonaldsonUhlenbeckYau on Hermitian YangMills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to bound- edness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of ε-Fano type. Furthermore, we show birational boundedness of 3-folds of ε-Fano type.