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 
MSC-Numbers: 53C30
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 tex2html_wrap_inline18 satisfying the condition tex2html_wrap_inline20. A special symplectic manifold is then defined as a special complex manifold together with a tex2html_wrap_inline18-parallel symplectic form tex2html_wrap_inline24. The Hodge components tex2html_wrap_inline26, tex2html_wrap_inline28, tex2html_wrap_inline30 are shown to be closed. If the form tex2html_wrap_inline32 is nondegenerate, it defines a (pseudo) Kähler metric tex2html_wrap_inline34 on M and if tex2html_wrap_inline32 is tex2html_wrap_inline18-parallel (e.g., if tex2html_wrap_inline42) then tex2html_wrap_inline44 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 tex2html_wrap_inline46. Locally, any special complex manifold is realised as the image of a local holomorphic 1-form tex2html_wrap_inline48. Such a realisation induces a canonical tex2html_wrap_inline18-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 tex2html_wrap_inline54. We include special complex manifolds tex2html_wrap_inline56 in a one-parameter family tex2html_wrap_inline58, and define projective versions of special complex, symplectic and Kähler manifolds in terms of an action of tex2html_wrap_inline60 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.

07.10.2017, 01:40