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Isometric Families of Kähler Structures

In: The Chern Symposium 1979

Author

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  • Eugenio Calabi

    (University of Pennsylvania, Department of Mathematics
    Institute for Advanced Study)

Abstract

The present study is motivated by the classical problem, loosely stated, of determining and describing, for any given family of complex analytic structures on a fixed, underlying real manifold, which subfamily consists of algebraic manifolds. This problem, which is probably inaccessible in its full generality, even when correctly stated with its missing qualifications, is treated here in a very restricted case, where each of the complex structures of the families under consideration admits a Kähler metric, such that the complex manifolds of the same family are isometric (though not complex analytically, in general). Clearly, this situation is highly restrictive compared to the general problem; however the information that one can extract from the two typical classes of such isometric families that can occur provides some interesting results on the global structure of some moduli spaces of nonalgebraic varieties, such as the K3-surfaces, the complex tori, and a special type of 2n-dimensional, rational affine variety.

Suggested Citation

  • Eugenio Calabi, 1980. "Isometric Families of Kähler Structures," Springer Books, in: W.-Y. Hsiang & S. Kobayashi & I. M. Singer & J. Wolf & H.-H. Wu & A. Weinstein (ed.), The Chern Symposium 1979, pages 23-39, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8109-9_3
    DOI: 10.1007/978-1-4613-8109-9_3
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