NDLI: Symplectic Dolbeault operators on Kähler manifolds

Content Provider	Springer Nature Link
Author	Korman, Eric O.
Copyright Year	2013
Abstract	For a Kähler manifold $$M$$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $$\bar{\partial }$$ and $$\bar{\partial }^*$$ , arise from Dirac operators on the canonical complex spinors on $$M$$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $$G/T$$ for $$G$$ a simply-connected compact semisimple Lie group and $$T$$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $$G$$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $$B_n$$ and $$C_n$$ . We give a thorough analysis of these operators on $$\mathbb{C } P^1$$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.
Ending Page	358
Page Count	20
Starting Page	339
File Format	PDF
ISSN	0232704X
e-ISSN	15729060
Journal	Annals of Global Analysis and Geometry
Issue Number	3
Volume Number	44
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-03-09
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Symplectic Dirac operators Flag manifolds Riemann surfaces Statistics for Business/Economics/Mathematical Finance/Insurance Analysis Theoretical, Mathematical and Computational Physics Kähler manifolds Group Theory and Generalizations
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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Erratum to: Symplectic Dolbeault operators on Kähler manifolds

Twistor Operators and Eigenvalues of the Dirac Operator on Compact Quaternion-Kähler Spin Manifolds

Locally homogeneous nearly Kähler manifolds

Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

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