Isometric Families of Kähler Structures
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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.
KeywordsComplex Manifold Abelian Variety Holonomy Group Hodge Decomposition Holomorphic Bundle
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- M. F. Atiyah, Some Examples of Complex Manifolds. Bonner Mathematische Schriften, No. 6, Bonn, 1958.Google Scholar
- R. B. Brown and A. Gray, Riemannian manifolds with holonomy group Spin(9), in Differential Geometry (Symposium in Honor of Kentaro Yano). Konokuniya, Tokyo, 1972, pp. 41–59.Google Scholar
- A. Lichnérowicz, Théorie Globale des Connexions et des Groupes d’Holonomie. Rome, Cremonese, 1955, esp. pp. 250–251.Google Scholar
- A. N. Todorov, Moduli of kählerian K-3 surfaces. Mathematische Arbeitstagung, Bonn, 1979.Google Scholar
- A. Weil, Théorèmes fondamentaux de la théorie des fonctions thêta (d’après des mémoires de Poincaré et Frobenius), in Sem. Bourbaki, No. 16, Mai 1949; Oeuvres scientifiques—Collected papers. Springer, New York, Heidelberg, New York, 1979, I, pp. 414–421.Google Scholar