On metric and cohomological properties of Oeljeklaus-Toma manifolds. (English) Zbl 1535.32025
The paper is focused on metric and cohomological features of Oeljeklaus-Toma manifolds. Oeljeklaus-Toma manifolds are generalisations of Inoue-Bombieri surfaces known for serving as counterexamples in various contexts.
Notably, Corollary 3 provides a comprehensive classification of Oeljeklaus-Toma manifolds equipped with pluriclosed metrics. Recall that a metric \(\omega\) over a complex manifold of dimension \(n\) is called pluriclosed (resp. astheno-Kähler, resp. strongly Gauduchon) if \(\partial \bar{\partial} \omega=0\) (resp. \(\partial \bar{\partial} \omega^{n-2}=0\), resp. \(\partial \omega^{n-1}\) is \(\bar{\partial}\)-exact). In Corollary 5 and Proposition 6 the authors show that Oeljeklaus-Toma manifolds never admit astheno-Kähler metrics or strongly Gauduchon metrics. On the cohomological side, they calculate the complex Bott-Chern cohomology of Oeljeklaus-Toma manifolds and their Betti and Hodge numbers when the manifolds admit a pluriclosed metric.
Notably, Corollary 3 provides a comprehensive classification of Oeljeklaus-Toma manifolds equipped with pluriclosed metrics. Recall that a metric \(\omega\) over a complex manifold of dimension \(n\) is called pluriclosed (resp. astheno-Kähler, resp. strongly Gauduchon) if \(\partial \bar{\partial} \omega=0\) (resp. \(\partial \bar{\partial} \omega^{n-2}=0\), resp. \(\partial \omega^{n-1}\) is \(\bar{\partial}\)-exact). In Corollary 5 and Proposition 6 the authors show that Oeljeklaus-Toma manifolds never admit astheno-Kähler metrics or strongly Gauduchon metrics. On the cohomological side, they calculate the complex Bott-Chern cohomology of Oeljeklaus-Toma manifolds and their Betti and Hodge numbers when the manifolds admit a pluriclosed metric.
Reviewer: Xiaojun Wu (Nice)
MSC:
| 32Q55 | Topological aspects of complex manifolds |
| 32Q15 | Kähler manifolds |
| 57T15 | Homology and cohomology of homogeneous spaces of Lie groups |
Keywords:
Bott-Chern cohomology; Hermitian metrics; Oeljeklaus-Toma manifolds; pluriclosed metrics; SKT metricsReferences:
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