Is There a Canonical Way of Giving a Complex n-Manifold a real 2n-str
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From: bacle on 9 Aug 2010 13:19 > On Aug 9, 1:34 pm, bacle <h...(a)here.com> wrote: > > Hi: > > Given a complex n-manifold M, i.e., complex > dimension n, > > > > is there a "natural", or "canonical" way of giving > M a > > > > real 2n-structure.? > > > > What degree of "smoothness" do you want? Thanks. I was hoping thatif M had a complex structure, then M with the real structure would automatically be C^oo . Isn't it.? Also: could you please suggest how to go in the opposite direction, i.e., given M a real, orientable, 2n-dimensional manifold: how would we give M a complex structure.?
From: bacle on 9 Aug 2010 19:49
> On Aug 10, 7:19 am, bacle <h...(a)here.com> wrote: > > > On Aug 9, 1:34 pm, bacle <h...(a)here.com> wrote: > > > > Hi: > > > > Given a complex n-manifold M, i.e., complex > > > dimension n, > > > > > > is there a "natural", or "canonical" way of > giving > > > M a > > > > > > real 2n-structure.? > > > > > What degree of "smoothness" do you want? > > > > Thanks. I was hoping thatif M had a complex > structure, > > then M with the real structure would > automatically be > > C^oo . Isn't it.? > > > > Yes, in fact it is even C^omega. > > > Also: could you please suggest how to go in > the opposite > > direction, i.e., given M a real, orientable, > 2n-dimensional > > manifold: how would we give M a complex > structure.? > > Do you understand what is meant by a C^omega > differentiable structure?. I think so; the coordinate-change functions are real-analytic, isn't it.?. I have been looking up results about extending real-analytic functions f on intervals (-r,r) ( work on (-r,r) for the sake of simplicity here) then f can be extended (or, more precisely, continued analytically) , into some domain containing the disk |z|=r , but I have not been able to generalize from this. Thanks. |