Introduction to Symplectic Geometry by Jean-Louis Koszul & Yi Ming Zou

Author:Jean-Louis Koszul & Yi Ming Zou
Language: eng
Format: epub
ISBN: 9789811339875
Publisher: Springer Singapore

be the canonical projections. Then is a symplectic structure on . We call the symplectic manifold the product of the symplectic manifolds and , and denote it by

2.1.5 Kähler structures. Let M be a 2n-dimensional manifold, and let J be a complex structure on M. As a tensor on M, J is of type (1, 1), thus we can view it as an endomorphism of the module of the vector fields on M. Then J satisfies the following conditions:(1), and

(2),

where X and Y are two arbitrary vector fields on M.

Let g be a differential symmetric 2-form on M such that holds for all vector fields X, Y on M. Let then .

If the rank of g is equal to 2n at every point of M, that is, g is a pseudo Riemannian form, then for any , is a symplectic form on the vector space and is a symplectic complex structure on . The form is a pseudo Hermitian form on M. If in addition , then is a symplectic structure on M, and h is called a pseudo Kähler form. Finally, if and g is positive definite on the tangent space at every point on M, then h is called a Kähler form on M. If h is a Kähler form, then for any , is a suitable complex structure on .

If h is a Kähler form on a complex manifold M, then for any complex submanifold N of M, the pullback of h on N is a Kähler form on N. In particular, all complex submanifolds of M are equipped with induced symplectic structures.

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