Abstract
The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, and the symmetrized polydisc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.
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Zhang, X., Ji, D. Umehara algebra and complex submanifolds of indefinite complex space forms. Ann Glob Anal Geom 63, 3 (2023). https://doi.org/10.1007/s10455-022-09876-8
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DOI: https://doi.org/10.1007/s10455-022-09876-8