Elliptic involutive structures and generalized Higgs algebroids

We study the module theory of two types of Lie algebroids: elliptic involutive structures (EIS) (which are equivalent to transversely holomorphic foliations) and what we call twisted generalized Higgs algebroids (TGHA). Generalizing the well-known results in the extremal cases of flat vector bundles and holomorphic vector bundles, we prove that there is an equivalence between modules over an EIS and locally free sheaves of modules over the sheaf of functions that are constant along the EIS. We define Atiyah like characteristic classes for such modules. Modules over a TGHA give a simultaneous generalization of Higgs bundles and generalized holomorphic vector bundles. For general Lie algebroids, we define a higher direct image construction of modules along a submersion. We also specialize to Higgs bundles, where we define Dolbeault cohomology valued secondary characteristic classes. We prove that these classes are compatible with the non-abelian Hodge theorem and the characteristic classes of flat vector bundles. We use these secondary classes to state and prove a refined Grothendieck-Riemann-Roch theorem for the pushforward of a Higgs bundle along a projection whose fiber is Kahler.