Hi, everyone: I am trying to show that any complex manifold is orientable.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Orientability of Complex Manifolds.

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