Abstract
This chapter summarizes the work of Calabi (Ann Math 58:1–23, 1953) about the existence of a Kähler immersion of a complex manifold into a finite or infinite dimensional complex space form . In particular, Calabi provides an algebraic criterion to find out whether a complex manifold admits or not such an immersion. Sections 2.1 and 2.2 are devoted to illustrate Calabi’s criterion for Kähler immersions into the complex Euclidean space and nonflat complex space forms respectively. In Sect. 2.3 we discuss the existence of a Kähler immersion of a complex space form into another, which Calabi himself in (Ann Math 58:1–23, 1953) completely classified as direct application of his criterion.
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References
S. Bochner, Curvature in Hermitian metric. Bull. Am. Math. Soc. 53, 179–195 (1947)
E. Calabi, Isometric imbedding of complex manifolds. Ann. Math. 58, 1–23 (1953)
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Loi, A., Zedda, M. (2018). Calabi’s Criterion. In: Kähler Immersions of Kähler Manifolds into Complex Space Forms. Lecture Notes of the Unione Matematica Italiana, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-99483-3_2
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DOI: https://doi.org/10.1007/978-3-319-99483-3_2
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