The Kähler Condition

Let be an almost complex manifold and suppose in addition we have a Riemannian metric such that is an orthogonal transformation with respect to . In symbols: For any vector fields and we have . In this case the triple is called a Hermitian Manifold.

Now we define a non-degenerate 2-form ( is called non-degenerate if ). This allows us to state the Kähler condition:


Definition 3.5

Suppose that is a Hermitian manifold if the associated 2-form is closed then we call a Kähler manifold.

 


For Kähler (and most generally symplectic) manifolds the existance of such an imposes topological restrictions on even-dimensional (orientable) manifolds admitting such structures.


Proposition 3.4

Let be a 2n-dimensional manifold with a closed non-degenerate 2-form . Then the even dimensional de Rham cohomology groups have strictly positive dimension.

 
Proof

The 2-form omega is non-degenerate, which is equivalent to the form being everywhere non-zero, and hence a volume form. If this form were d for an (n-1)-form then by Stoke's theorem the integral of over would be 0, hence this form represents a non-trivial cohomology class. Recall that the wedge map on forms descends to a product of cohomology groups so that , in particular we must have for each . We have a non-zero element of for .

 
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