  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 Levi-Civita connection replaced by Chern connection. We then
show that the tensor product of canonical and the anticanonical spinor bundles,
called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of
harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar
construction establishes isomorphisms between other types of harmonic spinors
and Bott-Chern, Aeppli and twisted cohomology.
