Linear complex structure

In mathematics, a complex structure on a real vector space V is is an real linear transformation

J : V → V

such that

J² = −id_V.

Here J² means J composed with itself and id_V is the identity map on V. That is, the effect of applying J twice is the same as multiplication by −1. This is reminiscent of multiplication by i. A complex structure allows one to give V the structure of a complex vector space. Complex scalar multiplication can be defined by

(x + i y)v = xv + yJ(v)

for all real numbers x,y and all vectors v in V. One can check that this does, in fact, give V the structure of a complex vector space which we denote (V, J).

If (V, J) has complex dimension n then V must have real dimension 2n. That is, V admits a complex structure only if it even-dimensional. It is not hard to see that every even-dimensional vector space admits a complex structure. (One can define J on pairs e,f of basis vectors by Je = f and Jf = −e and the extend by linearity to all of V).

A real linear transformation A : V → V is a complex linear transformation of the corresponding complex space (V, J) iff A commutes with J, i.e.

AJ = JA

Likewise, a real subspace U of V is a complex subspace of (V, J) iff J preserves U, i.e.

JU = U

Relation to complexifications

If J is a complex structure on V, we may extend J by linearity to the complexification of V,

$V^{C}=V\otimes _{\mathbb {R} }\mathbb {C} .$

Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ² = −1, namely λ = ±i. Thus we may write V^C = V⁺ ⊕ V⁻, where V⁺ and V⁻ are the eigenspaces of +i and −i, respectively. Complex conjugation provides a congugate-linear isomorphism over C between V⁺ and V⁻, and thus they have the same complex dimension. Thus if n is the complex dimension of V⁺, then 2n is the complex dimension of V^C, and so 2n is also the real dimension of V. Here V⁺ is the complex version of V that we defined above, while V⁻ is the complex space that results from defining J to be multiplication by −i.

Compatibility with other structures

If B is a bilinear form on V then we say that J preserves B if

B(Ju, Jv) = B(u, v)

for all u,v in V. An equivalent characterization is that J is skew-adjoint with respect to B:

B(Ju, v) = −B(u, Jv)

If g is an inner product on V then J preserves g iff J is an orthogonal transformation. Likewise, J preserves a nondegenerate, skew-symmetric form ω iff J is a symplectic transformation. For symplectic forms ω there is usually an added restriction for compatibility between J and ω, namely

ω(u, Ju) > 0

for all u in V. If this condition is satisfied then J is said to tame ω.

See also

• almost complex manifold