

Preprint 63/1999
Special complex manifolds
D. V. Alekseevsky, Vincente Cortés, and Chandrashekar Devchand
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Submission date: 12. Jan. 2000 (revised version: January 2000)
Pages: 24
published in: Journal of geometry and physics, 42 (2002) 1-2, p. 85-105
Bibtex
MSC-Numbers: 53C30
Keywords and phrases: special geometry, special kähler manifolds, hypercomplex manifolds, flat connections
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Abstract:
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.