Realisation of special Kähler manifolds as parabolic spheres

Baues, Oliver; Cortés, Vicente

doi:10.1090/S0002-9939-00-05981-5

Realisation of special Kähler manifolds as parabolic spheres
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by Oliver Baues and Vicente Cortés PDF

Proc. Amer. Math. Soc. 129 (2001), 2403-2407 Request permission

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

D.V. Alekseevsky, V. Cortés, C. Devchand, Special complex manifolds, math.DG/9910091, ESI preprint 779.
W. Blaschke, Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften VII, Springer Verlag, Berlin 1923.
Vicente Cortés, On hyper-Kähler manifolds associated to Lagrangian Kähler submanifolds of $T^*\textbf {C}^n$, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3193–3205. MR 1466946, DOI 10.1090/S0002-9947-98-02156-4
Daniel S. Freed, Special Kähler manifolds, Comm. Math. Phys. 203 (1999), no. 1, 31–52. MR 1695113, DOI 10.1007/s002200050604
Zhiqin Lu, A note on special Kähler manifolds, Math. Ann. 313 (1999), no. 4, 711–713. MR 1686939, DOI 10.1007/s002080050278
Katsumi Nomizu and Takeshi Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994. Geometry of affine immersions. MR 1311248