Abstract
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 $\mathcal{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 $\mathcal{V} $-spinor bundle and twisted cohomology.
Citation
Francesco Pediconi. "Chern-Dirac bundles on non-Kähler Hermitian manifolds." Rocky Mountain J. Math. 48 (4) 1255 - 1290, 2018. https://doi.org/10.1216/RMJ-2018-48-4-1255
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