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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spinors as automorphisms of the tangent bundle


Author: Alexandru Scorpan
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2049-2066
MSC (2000): Primary 53C27; Secondary 57N13, 32Q60, 53D05
Published electronically: December 12, 2003
MathSciNet review: 2031052
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Abstract: We show that, on a $4$-manifold $M$ endowed with a $\operatorname{spin}^{\mathbb{C} }$-structure induced by an almost-complex structure, a self-dual (positive) spinor field $\phi\in\Gamma(W^+)$ is the same as a bundle morphism $\phi:T_M\to T_M$ acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of $\phi$ on tangent vectors, and that the squaring map $\sigma:\mathcal{W}^+\to\Lambda^+$ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.


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Additional Information

Alexandru Scorpan
Affiliation: Department of Mathematics, University of California Berkeley, 970 Evans Hall, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
Email: scorpan@math.berkeley.edu, ascorpan@math.ufl.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03531-1
Keywords: Spinor, four-manifold, almost-complex, symplectic, K\"ahler
Received by editor(s): April 24, 2002
Received by editor(s) in revised form: April 15, 2003
Published electronically: December 12, 2003
Article copyright: © Copyright 2003 American Mathematical Society