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Special Kaehler manifolds: a survey. (English) Zbl 1039.53079
Slovák, Jan (ed.) et al., The proceedings of the 21th winter school “Geometry and physics”, Srní, Czech Republic, January 13–20, 2001. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 69, 11-18 (2002).
This is a survey of recent contributions to the area of special Kähler geometry. A (pseudo-)Kähler manifold \((M,J,g)\) is a differentiable manifold endowed with a complex structure \(J\) and a (pseudo-)Riemannian metric \(g\) such that i) \(J\) is orthogonal with respect to the metric \(g,\) ii) \(J\) is parallel with respect to the Levi Civita connection \(D.\) A special Kähler manifold \((M,J,g,\nabla)\) is a Kähler manifold \((M,J,g)\) together with a flat torsionfree connection \(\nabla\) such that i) \(\nabla \omega = 0,\) where \(\omega = g(.,J.)\) is the Kähler form and ii) \(\nabla\) is symmetric. A holomorphic immersion \(\phi : M \rightarrow V\) is called Kählerian if \(\phi^{\star} \gamma\) is nondegenerate and it is called Lagrangian if \(\phi^{\star}\Omega= 0.\)
Theorem 1. Let \(\phi:M \rightarrow V\) be a Kählerian Lagrangian immersion with induced geometric data \((g,\nabla).\) Then \((M,J,g,\nabla)\) is a special Kähler manifold. Conversely, any simply connected special Kähler manifold \((M,J,g,\nabla)\) admits a Kählerian Lagrangian immersion \(\phi : M \rightarrow V\) inducing the data \((g,\nabla)\) on \(M.\) The Kählerian Lagrangian immersion \(\phi\) is unique up to an affine transformation of \(V ={\mathbb C}^{2n}\) with linear part in Sp\(({\mathbb R}^{2n}).\) The hypersurface \(\varphi : M \rightarrow {\mathbb R}^{m+1}\) is called a parabolic hypersphere if the affine normal is parallel, \(\widetilde \nabla\xi = 0.\)
Theorem 2. Let \((M,J,g,\nabla)\) be a simply connected special Kähler manifold. Then there exists a parabolic hypersphere \(\varphi : M \longrightarrow {\mathbb R}^{m+1}, m =\dim _{{\mathbb R}}M = 2n,\) with Blaschke data \((g,\nabla).\) The immersion \(\varphi\) is unique up to unimodular affine transformation of \({\mathbb R}^{m+1}.\)
Theorem 4. If the Blaschke metric \(g\) of a parabolic affine hypersphere \((M,J, \nabla)\) is definite and complete, then \(M\) is affinely congruent to the paraboloid \(x^{m+1}= \sum^{m}_{i=1} (x^{i})^{2}\) in \({\mathbb R}^{m+1}.\) In particular, \(\nabla\) is the Levi Civita connection and \(g\) is flat.
For the entire collection see [Zbl 0994.00029].

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q15 Kähler manifolds
32Q20 Kähler-Einstein manifolds
32Q25 Calabi-Yau theory (complex-analytic aspects)
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry