MiS Preprint Repository

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection $\mathrm{\nabla }$ satisfying the condition $d^{\mathrm{\nabla }}J=0$. A special symplectic manifold is then defined as a special complex manifold together with a $\mathrm{\nabla }$-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 $g={\omega }^{11}oJ$ on M and if ${\omega }^{11}$ is $\mathrm{\nabla }$-parallel (e.g., if $\omega ={\omega }^{11}$) then $(M,J,\mathrm{\nabla },{\omega }^{11}$ 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 $T\ast C^n$. Locally, any special complex manifold is realised as the image of a local holomorphic 1-form $\xi :C^n\to T\ast C^n$. Such a realisation induces a canonical $\mathrm{\nabla }$-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 $(M,J,\mathrm{\nabla }$ in a one-parameter family $(M,J,{\mathrm{\nabla }}^{\vartheta }),\vartheta \in S^1$, and define projective versions of special complex, symplectic and Kähler manifolds in terms of an action of $C\ast$ 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.