MiS Preprint Repository

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection $\nabla $ satisfying the condition $d^\nabla J =0$. A special symplectic manifold is then defined as a special complex manifold together with a $\nabla$-parallel symplectic form $\omega$. The Hodge components $\omega^{11},\omega^{20}, \omega^{02} $ are shown to be closed. If the form $\omega^{11}$ is nondegenerate, it defines a (pseudo) Kähler metric $g=\omega^{11} o J$ on M and if $\omega^{11}$ is $\nabla$-parallel (e.g., if $\omega = \omega^{11}$) then $(M,J,\nabla,\omega^{11}$ is a special Kähler manifold in the sense of Freed. We give an extrinsic realisation of simply connected special complex, symplectic and Kähler manifolds as immersed complex submanifolds of $T*C^n$. Locally, any special complex manifold is realised as the image of a local holomorphic 1-form $\xi : C^n \to T*C^n $. Such a realisation induces a canonical $\nabla$-parallel symplectic structure on M and any special symplectic manifold is locally obtained this way. Special Kähler manifolds are realised by complex Lagrangian submanifolds and correspond to closed forms $\xi$. We include special complex manifolds $(M,J,\nabla$ in a one-parameter family $(M,J,\nabla^\vartheta), \vartheta \in S^1$, and define projective versions of special complex, symplectic and Kähler manifolds in terms of an action of $C*$ on M which is transitive on this family. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which are generalisations of the known hyper-Kähler structure on the cotangent bundle of a special Kähler manifold.