Avoid common mistakes on your manuscript.
1 Erratum to: Ann Glob Anal Geom DOI 10.1007/s10455-013-9369-x
The Hermitian structure used for a flag manifold \(G/T\) in Sect. 4 is not a Kähler structure unless \(G\) is a product of \(SU(2)\)’s. This is because the connection used is the canonical connection on the homogeneous space \(G/T\), which has non-zero torsion since \([\mathfrak{t }^\perp , \mathfrak{t }^\perp ] \not \subseteq \mathfrak{t }\). Indeed, using the metric to lower indices, the torsion at \(eT\) is the 3-form given by
Such a structure on \(G/T\) turns it into what is called a “Kähler with torsion (KT)” manifold. The general results and constructions established in the previous sections carry over unchanged to KT manifolds except for Proposition 3 which states that the symplectic Dirac operators \(D\) and \(\tilde{D}\) are formally self-adjoint. For connections with torsion and parallel complex structure, a sufficient condition for \(D\) and \(\tilde{D}\) to be self-adjoint is the vanishing of the torsion vector field, defined by
where \(\mathbf{T}\) is the torsion of \(\nabla \) and \(\{a_1,\ldots ,a_n, b_1,\ldots ,b_n\}\) is a symplectic frame [2]. In the case of flag manifolds, a symplectic basis at \(eT\) is proportional to \(\{Z_\alpha , Z_{-\alpha }\}\) where \(Z_\alpha \) is a root vector for \(\alpha \). Since \(\mathbf{T}(Z_\alpha , Z_{-\alpha }) = -[Z_\alpha , Z_{-\alpha }]_{\mathfrak{t }^\perp } = 0\), we see that \(\mathcal{T }\) vanishes. Thus the results concerning flag manifolds are correct using the symplectic Dolbeault operators corresponding to their KT structures.
References
Kobayashi, S., Nomizum, K.: Foundations of Differential Geometry, vol. 2, pp. 192–193. Interscience Publishers, New York (1969)
Habermann, K., Habermann, L.: Introduction to symplectic Dirac operators. In: Lecture Notes in Mathematics. Springer, Berlin (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
The online version of the original article can be found under doi:10.1007/s10455-013-9369-x.
Rights and permissions
About this article
Cite this article
Korman, E.O. Erratum to: Symplectic Dolbeault operators on Kähler manifolds. Ann Glob Anal Geom 44, 359–360 (2013). https://doi.org/10.1007/s10455-013-9384-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-013-9384-y