Abstract
Given a quaternionic manifold $M$ with a certain $\mathrm{U}(1)$-symmetry, we construct a hypercomplex manifold $M'$ of the same dimension. This construction generalizes the quaternionic Kähler/ hyper-Kähler-correspondence. As an example of this construction, we obtain a compact homogeneous hypercomplex manifold which does not admit any hyper-Kähler structure. Therefore our construction is a proper generalization of the quaternionic Kähler/hyper-Kähler-correspondence.
Acknowledgments
We thank Aleksandra Borówka and Henrik Winther for discussions. Research by the first author is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2121 Quantum Universe - 390833306. This paper was prepared during the second author's stay at Universität Hamburg on his sabbatical leave (April 2018 - March 2019). He would like to thank the Department of Mathematics of Universität Hamburg for the hospitality. His research is partially supported by JSPS KAKENHI Grant Number 18K03272.
Citation
Vicente Cortés. Kazuyuki Hasegawa. "The quaternionic/hypercomplex-correspondence." Osaka J. Math. 58 (1) 213 - 238, January 2021.
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