Quaternion-K��hler manifolds near maximal fixed points sets of $S^1$-symmetries
| Autor: | Bor��wka, Aleksandra |
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| Rok vydání: | 2019 |
| Předmět: | |
| DOI: | 10.48550/arxiv.1904.08474 |
| Popis: | Using quaternionic Feix--Kaledin construction we provide a local classification of quaternion-K��hler metrics with a rotating $S^1$-symmetry with the fixed point set submanifold $S$ of maximal possible dimension. For any K��hler manifold $S$ equipped with a line bundle with a unitary connection of curvature proportional to the K��hler form we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix-Kaledin construction from this data. Conversely, we show that quaternion-K��hler metrics with a rotating $S^1$-symmetry induce on the fixed point set of maximal dimension a K��hler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the K��hler form and the two constructions are inverse to each other. Moreover, we study the case when $S$ is compact, showing that in this case the quaternion-K��hler geometry is determined by the K��hler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle. Finally, we relate the results to the c-map construction showing that the family of quaternion-K��hler manifolds obtained from a fixed K��hler metric on $S$ by varying the line bundle and the hyperk��hler manifold obtained by hyperk��hler Feix--Kaledin construction form $S$ are related by hyperk��hler/quaternion-K��hler correspondence. 16 pages |
| Databáze: | OpenAIRE |
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