Birational Functors in the Derived Category

Abstract

In this thesis, we study a class of derived equivalences that naturally induce birational maps. We give several equivalent criteria for a birational correspondence to exist, and prove the correspondence induces a $K$-equivalece, extending a result of Kawamata. With the introduction of a canonical natural transformation from the right to left adjoint, we are also able to extend this study to the fully faithful situation. This natural transformation gives us context to provide a new proof of Bridgland's equivalence criterion and of the indecomposability of the derived category in the trivial canonical bundle. Additionally, we construct an algebraic moduli space of birational integral transforms inside of Lieblich's moduli of complexes.