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

  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. 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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