Derived Equivalences of Irregular Varieties and Constraints on Hodge Numbers

We study derived equivalences of smooth projective irregular varieties. More specifically, as suggested by a conjecture of Popa, we investigate the behavior of cohomological support loci associated to the canonical bundle (around the origin) under derived equivalence. We approach this problem in two ways. In the first approach we establish and apply the derived invariance of a ``twisted'' version of Hochschild homology taking into account an isomorphism due to Rouquier and related to autoequivalences of derived categories. In the second approach we relate the derived invariance of cohomological support loci to the derived invariance of Hodge numbers. As a result, we obtain the derived invariance of the first two and the last two cohomological support loci, leading to interesting geometric applications. For instance, we deduce the derived invariance of a few numerical quantities attached to irregular varieties, and furthermore we describe the geometry of Fourier-Mukai partners of Fano fibrations, and hence of Mori fiber spaces, fibered over curves of genus at least two. Finally, we also study constraints on Hodge numbers of special classes of irregular compact Kaehler manifolds. More specifically, we write down nequalities for all the Hodge numbers by studying the exactness of BGG complexes associated to bundles of holomorphic p-forms and by using classical results in the theory of vector bundles on projective spaces. As an application of our techniques, we bound the regularity of cohomology modules in terms of the defect of semismallness of the Albanese map.