Abstract
Let \((M,J,g,\omega )\) be a complete Hermitian manifold of complex dimension \(n\ge 2\). Let \(1\le p\le n-1\) and assume that \(\omega ^{n-p}\) is \((\partial +\overline{\partial })\)-bounded. We prove that, if \(\psi \) is an \(L^2\) and d-closed (p, 0)-form on M, then \(\psi =0\). In particular, if M is compact, we derive that if the Aeppli class of \(\omega ^{n-p}\) vanishes, then \(H^{p,0}_{BC}(M)=0\). As a special case, if M admits a Gauduchon metric \(\omega \) such that the Aeppli class of \(\omega ^{n-1}\) vanishes, then \(H^{1,0}_{BC}(M)=0\).
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Acknowledgements
We are grateful to Daniele Angella, Paul Gauduchon, Nicoletta Tardini and Valentino Tosatti for useful conversations and helpful comments. We also thank the anonymous referee for all the suggestions for a better presentation of the work.
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Communicated by Filippo Bracci.
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This work was partially supported by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, Project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by GNSAGA of INdAM.
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Piovani, R., Tomassini, A. Aeppli Cohomology and Gauduchon Metrics. Complex Anal. Oper. Theory 14, 22 (2020). https://doi.org/10.1007/s11785-020-00984-6
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DOI: https://doi.org/10.1007/s11785-020-00984-6