TITLE: What is the difference between real and complex manifolds?
QUESTION [17 upvotes]: Given an even-dimensional (smooth) manifold, what is the difference between its (real) smooth structure and its complex structure? I realize that in the real case, the overlap functions of charts need only be smooth, while in the complex case they need to be holomorphic. However, this doesn't provide a satisfactory answer- it begs the question of why holomorphicity is a stronger condition than smoothness in the first case. The answer to which is just that "in the real case, limits can only go from either side, whereas in the complex case they can be taken from all sorts of directions"- but this doesn't not seem very rigorous. Any thoughts on what is really at work here?

REPLY [12 votes]: There's a sort of "standard" answer to this question, involving the notion of an almost complex manifold.
A complex manifold has the property that each tangent space is endowed with the structure of a complex vector space.  Roughly speaking, the difference between a real vector space of dimension $2n$ and a complex vector space of dimension $n$ is that in the complex vector space, you know what it means to multiply by $i$.  More formally, you can put a complex structure on any real vector space by choosing any linear transformation $J$ such that $J^2 = -1$, with $J$ serving as multiplication by $i$.
Note that this $J$ adds a lot of structure to a real vector space.  For example, each vector $v$ now has an associated $2$-dimensional subspace, namely the span of $v$ and $Jv$ (i.e. the set of complex multiples of $v$), and it's possible to measure angles between vectors on this subspace.  Also, we can now classify linear transformations of the vector space into two types -- those that commute with $J$ (i.e. complex linear transformations) and those that don't.
A complex manifold of dimension $n$ can be thought of as a real manifold of dimension $2n$ together with a smoothly varying choice of $J$ for each tangent space.  Indeed, a real manifold endowed with such a smooth choice of $J$'s is called an almost complex manifold.
The word "almost" is pertinent because not every smooth choice of $J$'s actually yields a complex manifold.  You have to choose a set of $J$'s that are "integrable" in the sense that a certain tensor defined in terms of $J$ vanishes.  My impression is that the geometric difference between almost complex manifolds and complex manifolds is not well-understood.