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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?

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    Another difference is that the complex structure gives the manifold a canonical orientation whereas there are lots of real even-dimensional manifolds that aren't orientable at all.2011-11-30
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    VERY good question that it's hard to get a straight answer to.Believe me,I've tried to get one and failed.2011-11-30

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