Linear Complex Structure - Relation To Complexifications

Relation To Complexifications

Given any real vector space V we may define its complexification by extension of scalars:

This is a complex vector space whose complex dimension is equal to the real dimension of V. It has a canonical complex conjugation defined by

If J is a complex structure on V, we may extend J by linearity to VC:

Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ2 = −1, namely λ = ±i. Thus we may write

where V+ and V− are the eigenspaces of +i and −i, respectively. Complex conjugation interchanges V+ and V−. The projection maps onto the V± eigenspaces are given by

So that

There is a natural complex linear isomorphism between V_J and V+, so these vector spaces can be considered the same, while V− may be regarded as the complex conjugate of V_J.

Note that if V_J has complex dimension n then both V+ and V− have complex dimension n while VC has complex dimension 2n.

Abstractly, if one starts with a complex vector space W and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of W and its conjugate: