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A new positivity condition for the curvature of Hermitian manifolds

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Abstract

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kähler case. We derive a Bochner formula for closed (1, 1)-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfies our positivity condition, then any class \(\alpha \in H^{1, 1}_{BC}(X)\) can be represented by a closed (1, 1)-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

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Acknowledgements

I would like to thank my advisor Duong Phong for many helpful suggestions and for his constant support and encouragement. I am also grateful to Nikita Klemyatin for bringing to my attention the paper [19].

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Correspondence to Freid Tong.

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This work is supported in part by NSF grant DMS-1855947.

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Tong, F. A new positivity condition for the curvature of Hermitian manifolds. Math. Z. 298, 1175–1185 (2021). https://doi.org/10.1007/s00209-020-02647-w

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