Abstract
Let X be a projective manifold and let {θ}∈ H1,1(X, ℝ) be a nonzero pseudo-effective (transcendental) class, where θ is a smooth closed real (1, 1)-form. We prove that if for any one-dimensional complex submanifold C ⊂ X and ϕ ∈ SPsh(C, θ∣C) with a single analytic singularity at some point p ∈ C, there exists a function \(\tilde \varphi \in {\rm{Psh}}(X,\theta )\) such that \(\tilde \varphi \left| {_C = \varphi } \right.\) and \({\tilde \varphi }\) is continuous at points of C {p}, then {θ} is a Kähler class.
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Acknowledgements
The first author was supported by National Natural Science Foundation of China (Grant No. 11901046). The second author was supported by the Beijing Natural Science Foundation (Grant Nos. 1202012 and Z190003) and National Natural Science Foundation of China (Grant No. 12071035). This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFA1002600). The authors thank Professor Xiangyu Zhou for his constant support and guidance. The authors are grateful to the referees for their careful reading and for their suggestions which helped to clarify the exposition.
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Meng, X., Wang, Z. A Kählerness criterion for real (1, 1)-classes on projective manifolds through extendibility of singular potentials. Sci. China Math. 65, 1795–1802 (2022). https://doi.org/10.1007/s11425-021-1934-1
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DOI: https://doi.org/10.1007/s11425-021-1934-1