Hermitian structures on a class of quaternionic Kähler manifolds
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909384418787328 |
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| author | Cortés, V. Saha, A. Thung, D. |
| author_facet | Cortés, V. Saha, A. Thung, D. |
| contents | Any quaternionic Kähler manifold $(\bar N,g_{\bar N},\mathcal Q)$ equipped with a Killing vector field $X$ with nowhere vanishing quaternionic moment map carries an integrable almost complex structure $J_1$ that is a section of the quaternionic structure $\mathcal Q$. Using the HK/QK correspondence, we study properties of the almost Hermitian structure $(g_{\bar N},\tilde J_1)$ obtained by changing the sign of $J_1$ on the distribution spanned by $X$ and $J_1X$. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally Kähler. We show that for a large class of quaternionic Kähler manifolds containing the one-loop deformed c-map spaces, the structure $\tilde J_1$ is integrable. We do also show that the integrability of $\tilde J_1$ implies that $(g_{\bar N},\tilde J_1)$ is conformally Kähler in dimension four, but not in higher dimensions. In the special case of the one-loop deformation of the quaternionic Kähler symmetric spaces dual to the complex Grassmannians of two-planes we construct a third canonical Hermitian structure $(g_{\bar N},\hat J_1)$. Finally, we give a complete local classification of quaternionic Kähler four-folds for which $\tilde J_1$ is integrable and show that these are either locally symmetric or carry a cohomogeneity $1$ isometric action generated by one of the Lie algebras $\mathfrak{o}(2)\ltimes\mathfrak{heis}_3(\mathbb R)$, $\mathfrak{u}(2)$, or $\mathfrak{u}(1,1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_07946 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hermitian structures on a class of quaternionic Kähler manifolds Cortés, V. Saha, A. Thung, D. Differential Geometry 53C26 Any quaternionic Kähler manifold $(\bar N,g_{\bar N},\mathcal Q)$ equipped with a Killing vector field $X$ with nowhere vanishing quaternionic moment map carries an integrable almost complex structure $J_1$ that is a section of the quaternionic structure $\mathcal Q$. Using the HK/QK correspondence, we study properties of the almost Hermitian structure $(g_{\bar N},\tilde J_1)$ obtained by changing the sign of $J_1$ on the distribution spanned by $X$ and $J_1X$. In particular, we derive necessary and sufficient conditions for its integrability and for it being conformally Kähler. We show that for a large class of quaternionic Kähler manifolds containing the one-loop deformed c-map spaces, the structure $\tilde J_1$ is integrable. We do also show that the integrability of $\tilde J_1$ implies that $(g_{\bar N},\tilde J_1)$ is conformally Kähler in dimension four, but not in higher dimensions. In the special case of the one-loop deformation of the quaternionic Kähler symmetric spaces dual to the complex Grassmannians of two-planes we construct a third canonical Hermitian structure $(g_{\bar N},\hat J_1)$. Finally, we give a complete local classification of quaternionic Kähler four-folds for which $\tilde J_1$ is integrable and show that these are either locally symmetric or carry a cohomogeneity $1$ isometric action generated by one of the Lie algebras $\mathfrak{o}(2)\ltimes\mathfrak{heis}_3(\mathbb R)$, $\mathfrak{u}(2)$, or $\mathfrak{u}(1,1)$. |
| title | Hermitian structures on a class of quaternionic Kähler manifolds |
| topic | Differential Geometry 53C26 |
| url | https://arxiv.org/abs/2206.07946 |