Kahler and symplectic structures on 4-manifolds and hyperKahler geometry
Abstract
A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifold. For a class of hyperKahler manifolds obtained via hyperKahler reduction, we also show that a harmonic spinor, under mild conditions, defines a symplectic structure. Finally, we show that if a covariantly constant, generalized spinor satisfies generalized Seiberg-Witten equations, the metric on the base manifold has a constant scalar curvature.
Keywords
Cite
@article{arxiv.1606.08158,
title = {Kahler and symplectic structures on 4-manifolds and hyperKahler geometry},
author = {Varun Thakre},
journal= {arXiv preprint arXiv:1606.08158},
year = {2016}
}
Comments
15 pages, added the case of hyperKahler manifolds obtained via hyperKahler reduction and rectified a couple of embarrassing mistakes in the previous version of the paper. Comments and suggestions are welcome!