# zbMATH — the first resource for mathematics

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].

##### MSC:
 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