Maximal Exterior product of Tangent space

[nughret]

I am working through a book on Kahler manifolds and for one of the proofs it states that the maximum exterior power of TM is m (where M has complex dimension). Could you explain why this is the case rather than the maximum exterior power being 2m.

 

[wofsy]

I am working through a book on Kahler manifolds and for one of the proofs it states that the maximum exterior power of TM is m (where M has complex dimension). Could you explain why this is the case rather than the maximum exterior power being 2m.

My guess is that you are wedging even dimensional forms.

 

[zhentil]

The torus seems to provide a counterexample, absent further information.