Special complex manifolds
D. V. Alekseevsky, Vincente Cortés,and Chandrashekar Devchand
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Submission date: 12. Jan. 2000 (revised version: January 2000)
published in: Journal of geometry and physics, 42 (2002) 1-2, p. 85-105
Keywords and phrases: special geometry, special kähler manifolds, hypercomplex manifolds, flat connections
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We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection satisfying the condition . A special symplectic manifold is then defined as a special complex manifold together with a -parallel symplectic form . The Hodge components , , are shown to be closed. If the form is nondegenerate, it defines a (pseudo) Kähler metric on M and if is -parallel (e.g., if ) then 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 . Locally, any special complex manifold is realised as the image of a local holomorphic 1-form . Such a realisation induces a canonical -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 . We include special complex manifolds in a one-parameter family , and define projective versions of special complex, symplectic and Kähler manifolds in terms of an action of 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.