## Rocky Mountain Journal of Mathematics

### Chern-Dirac bundles on non-Kähler Hermitian manifolds

Francesco Pediconi

#### 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.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1255-1290.

Dates
First available in Project Euclid: 30 September 2018

https://projecteuclid.org/euclid.rmjm/1538272833

Digital Object Identifier
doi:10.1216/RMJ-2018-48-4-1255

Mathematical Reviews number (MathSciNet)
MR3859758

Zentralblatt MATH identifier
06958779

#### Citation

Pediconi, Francesco. Chern-Dirac bundles on non-Kähler Hermitian manifolds. Rocky Mountain J. Math. 48 (2018), no. 4, 1255--1290. doi:10.1216/RMJ-2018-48-4-1255. https://projecteuclid.org/euclid.rmjm/1538272833

#### References

• I. Agricola, The Srní lectures on non-integrable geometries with torsion, Arch. Math. 42 (2006), 5–84.
• D. Angella and A. Tomassini, On the $\partial \bar{\partial}$-lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), 71–81.
• ––––, On Bott-Chern cohomology and formality, J. Geom. Phys. 93 (2015), 52–61.
• V. Apostolov and G. Dloussky, Locally conformally symplectic structures on compact non-Kähler complex surfaces, Int. Math. Res. 9 (2016), 2717–2747.
• T. Friedrich, Dirac operators in Riemannian geometry, American Mathematical Society, Providence, RI, 2000.
• P. Gauduchon, Hermitian connections and Dirac operators, Boll. UMI 11 (1997), 257–288.
• N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55.
• H.B. Lawson, Jr., and M.L. Michelsohn, Spin geometry, Princeton University Press, Princeton, 1989.
• A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris 257 (1963), 7–9.
• M.L. Michelsohn, Clifford and spinor cohomology of Kähler manifolds, Amer. J. Math. 102 (1980), 1083–1146.