Hi, everyone: I am trying to show that any complex manifold is orientable. I know this has to see with properties of Gl(n;C) (C complexes, of course.) ; specifically, with Gl(n;C) being connected (as a Lie Group.). Now this means that the determinant map must be either always pos. or always negative, but I am not clear on why it is not always negative. Also, I am confused about the fact that the determinant may be complex-valued, so that it does not make sense to say it is positive or negative. Any Ideas.? Thanks.