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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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