  We study the behavior of cohomological support loci of the canonical bundle
under derived equivalence of smooth projective varieties. This is achieved by
investigating the derived invariance of a generalized version of Hochschild
homology. Furthermore, using techniques coming from birational geometry, we
establish the derived invariance of the Albanese dimension for varieties having
non-negative Kodaira dimension. We apply our machinery to study the derived
invariance of the holomorphic Euler characteristic and of certain Hodge numbers
for special classes of varieties. Further applications concern the behavior of
particular types of fibrations under derived equivalence.
