Abstract
Let \(p:X\rightarrow Y\) be a surjective holomorphic mapping between Kähler manifolds. Let D be a smoothly bounded domain in X such that every generic fiber \(D_y:=D\cap p^{-1}(y)\) for \(y\in Y\) is a strongly pseudoconvex domain in \(X_y:=p^{-1}(y)\), which admits the complete Kähler–Einstein metric. This family of Kähler–Einstein metrics induces a smooth (1, 1)-form \(\rho \) on D. In this paper, we prove that \(\rho \) is positive-definite on D if D is strongly pseudoconvex. We also discuss the extension of \(\rho \) as a positive current across singular fibers.
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Acknowledgements
The authors would like to thank G. Schumacher and M. Păun for their valuable comments and suggestions. The first author was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea government (No. 2018R1C1B3005963). The second author was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea government (No. 2010-0020413).
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Choi, YJ., Yoo, S. Holomorphic Families of Strongly Pseudoconvex Domains in a Kähler Manifold. J Geom Anal 31, 2639–2655 (2021). https://doi.org/10.1007/s12220-020-00369-3
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DOI: https://doi.org/10.1007/s12220-020-00369-3