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Is GL(n,C) the biggest complex manifold within GL(2n,R)?

  1. Apr 2, 2014 #1
    So the question is simple: is [itex]GL_n(\mathbb C)[/itex] the biggest complex manifold in [itex]GL_{2n}(\mathbb R)[/itex]? Or is there another one that strictly contains it?
  2. jcsd
  3. Apr 6, 2014 #2
    I'm not sure this question makes sense. First of all, you don't mean biggest, you mean to ask if it's maximal, which is still a fairly weird question. I mean, a complex manifold of (complex) dimension n union C^n is still a complex manifold.

    As long is there is an open ball left in the complement, you can put a complex structure on that open ball, so you can conclude that any maximal complex submanifold would have to be dense in GL_2n(R) and therefore equal to it. I don't know whether GL_2n(R) can accommodate a complex structure, but this argument shows that there is never any such thing as a maximal complex submanifold of any manifold, unless it happens to equal the manifold itself. Not all manifolds have a complex structure, so if they don't, then what you would get is just bigger and bigger complex submanifolds containing any given one.
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