Special complex manifolds

Abstract

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition of (affine) special Kähler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and Kähler manifolds. We prove that the above three types of special geometry are completely solvable, in the sense that they are locally defined by free holomorphic data. In fact, any special complex manifold is locally realised as the image of a holomorphic 1-form α: 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 as complex Lagrangian submanifolds and correspond to closed forms α. Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which generalise the hyper-Kähler structure on the cotangent bundle of a special Kähler manifold.