Special Kaehler manifolds: A survey

summary:This is a survey of recent contributions to the area of special K\"ahler geometry. A (pseudo-)K\"ahler 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 K\"ahler manifold $(M,J,g,\nabla)$ is a K\"ahler manifold $(M,J,g)$ together with a flat torsionfree connection $\nabla$ such that i) $\nabla \omega = 0,$ where $\omega = g(.,J.)$ is the K\"ahler form and ii) $\nabla$ is symmetric. A holomorphic immersion $\phi : M \rightarrow V$ is called K\"ahlerian if $\phi^{\star} \gamma$ is nondegenerate and it is called Lagrangian if $\phi^{\star}\Omega= 0.$\par Theorem 1. Let $\phi:M \rightarrow V$ be a K\"ahlerian Lagrangian immersion with induced geometric data $(g,\nabla).$ Then $(M,J,g,\nabla)$ is a special K\"ahler manifold. Conversely, any simply connected sp