## Chern-Dirac bundles on non-Kähler Hermitian manifolds.(English)Zbl 1406.53054

Let $$(M,g,J)$$ be a Hermitian manifold, i.e., a complex manifold $$M$$ endowed with a Hermitian inner product $$g$$ on the tangent bundle, and $$J$$ is the endomorphism of the tangent bundle given by multiplication by $$i$$. Note that if $$\omega := g(J-,-)$$ is closed, then $$(M,g,J)$$ is a Kähler manifold.
The Chern connection of $$(M,g,J)$$ is the $$\mathfrak{u}_n$$-valued connection form $$\omega^\mathcal{C}$$ on the $$U_n$$-bundle $$U_{g,J}(M)$$ of $$(g,J)$$-unitary frames, whose associated covariant derivative $$D^\mathcal{C}$$ on vector fields of $$M$$ possesses torsion $$T$$ satisfying $$T(J-,-) = T(-,J-)$$. Note that the Chern connection coincides with the Levi-Civita connection if $$(M,g,J)$$ is Kähler.
A Chern-Dirac bundle over $$(M,g,J)$$ is similarly defined as a Dirac bundle, but with the Levi-Civita connection occuring in the definition replaced by the Chern connection. If $$E$$ is a Chern-Dirac bundle, then it is shown how to define the corresponding Chern-Dirac operator $$D$$ acting on it, which is a first-order, symmetric and elliptic operator. The first results derived in the paper for these Chern-Dirac operators are Bochner-type formulas.
The author then constructs on a given Hermitian manifold $$(M,g,J)$$ the $$\mathcal{V}$$-spinor bundle $$\mathcal{V}M$$ and gives a thorough treatment of it. Especially, there are two Chern-Dirac operators $$D^L$$ and $$D^R$$ defined on it, and the following result is proven: $\mathrm{ker}(D^L + D^R) \cong \bigoplus_{k=0}^{2n} H^k_{\mathrm{dR}}(M;\mathbb{C}) \quad\text{ and }\quad \mathrm{ker}(D^R) \cong \bigoplus_{p,q=0}^n H^{p,q}_{\bar\partial}(M),$ where $$H^\ast_{\mathrm{dR}}(M;\mathbb{C})$$ denotes de Rham cohomology and $$H^{\ast,\ast}_{\bar\partial}(M)$$ Dolbeault cohomology.

### MSC:

 53C27 Spin and Spin$${}^c$$ geometry 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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### References:

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