Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Realisation of special Kähler manifolds as parabolic spheres

Author(s): Oliver Baues; Vicente Cortés
Journal: Proc. Amer. Math. Soc. 129 (2001), 2403-2407.
MSC (2000): Primary 53A15, 53C26
Posted: November 30, 2000
MathSciNet review: 1823925
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Abstract:

We prove that any simply connected special Kähler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. We also show that a classical result of Calabi and Pogorelov on parabolic spheres implies Lu's theorem on complete special Kähler manifolds with a positive definite metric.

References:

[ACD]: D.V. Alekseevsky, V. Cortés, C. Devchand, Special complex manifolds, math.DG/9910091, ESI preprint 779.
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[C]: V. Cortés, On hyper-Kähler manifolds associated to Lagrangian Kähler submanifolds of $T\sp *{C}\sp n$ , Trans. Amer. Math. Soc. 350 (1998), no. 8, 3193-3205. MR 98k:53059
[F]: D.S. Freed, Special Kähler manifolds, Comm. Math. Phys. 203 (1999), no. 1, 31-52. MR 2000f:53060
[L]: Zhiqin Lu, A note on special Kähler manifolds, Math. Ann. 313 (1999), no. 4, 711-713. MR 2000f:53061
[NS]: K. Nomizu, T. Sasaki, Affine differential geometry. Geometry of affine immersions, Cambridge Tracts in Mathematics 111, Cambridge University Press, Cambridge, 1994. MR 96e:53014

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