Almost Complex Manifold - Formal Definition

Formal Definition

Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field J of rank (1,1) such that J 2 = −1 when regarded as a vector bundle isomorphism J : TM → TM on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold.

If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose M is n-dimensional, and let J : TM → TM be an almost complex structure. Then det(J-xI) is a polynomial in x of degree n. If n is odd, then it has a real root, z. Then det(J − zI) = 0, so there exists a vector v in TM with Jv = zv. Hence JJv = z2v which cannot equal −v if z is real. Thus n must be even if M has an almost complex structure. One can show that it must be orientable as well.

An easy exercise in linear algebra shows that any even dimensional vector space admits a linear complex structure. Therefore an even dimensional manifold always admits a (1, 1) rank tensor pointwise (which is just a linear transformation on each tangent space) such that J_p2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold M is equivalent to a reduction of the structure group of the tangent bundle from GL(2n, R) to GL(n, C). The existence question is then a purely algebraic topological one and is fairly well understood.