Chern-Dirac bundles on non-kahler hermitian manifolds

We introduce the notions of Chern--Dirac bundles and Chern--Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with the Levi-Civita connection replaced by the Chern connection. We then show that the tensor product of the canonical and the anticanonical spinor bundles, called the V-spinor bundle, is a bigraded Chern--Dirac bundle with spaces of harmonic sections isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms among other types of harmonic sections of the V-spinor bundle and twisted cohomology.