## Spinors as automorphisms of the tangent bundle

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- by Alexandru Scorpan PDF
- Trans. Amer. Math. Soc.
**356**(2004), 2049-2066 Request permission

## 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.## References

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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
- Received by editor(s): April 24, 2002
- Received by editor(s) in revised form: April 15, 2003
- Published electronically: December 12, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 2049-2066 - MSC (2000): Primary 53C27; Secondary 57N13, 32Q60, 53D05
- DOI: https://doi.org/10.1090/S0002-9947-03-03531-1
- MathSciNet review: 2031052