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.


53C27 Spin and Spin\({}^c\) geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: DOI arXiv Euclid


[1] I. Agricola, The Srní lectures on non-integrable geometries with torsion, Arch. Math. 42 (2006), 5–84. · Zbl 1164.53300
[2] D. Angella and A. Tomassini, On the \(∂ \bar{∂}\)-lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), 71–81. · Zbl 1271.32011
[3] ——–, On Bott-Chern cohomology and formality, J. Geom. Phys. 93 (2015), 52–61.
[4] V. Apostolov and G. Dloussky, Locally conformally symplectic structures on compact non-Kähler complex surfaces, Int. Math. Res. 9 (2016), 2717–2747.
[5] T. Friedrich, Dirac operators in Riemannian geometry, American Mathematical Society, Providence, RI, 2000. · Zbl 0949.58032
[6] P. Gauduchon, Hermitian connections and Dirac operators, Boll. UMI 11 (1997), 257–288. · Zbl 0876.53015
[7] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55. · Zbl 0284.58016
[8] H.B. Lawson, Jr., and M.L. Michelsohn, Spin geometry, Princeton University Press, Princeton, 1989. · Zbl 0688.57001
[9] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7–9. · Zbl 0136.18401
[10] M.L. Michelsohn, Clifford and spinor cohomology of Kähler manifolds, Amer. J. Math. 102 (1980), 1083–1146. · Zbl 0462.53035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.