Hi, everyone: I was finally able to show that all complex manifolds are orientable, by generalizing to many variables the fact that , for a single complex variable, the Jacobian matrix is of the form (after using Cauchy-Riemann to substitute). ( This is my first post here. I read the FAQ's, but I apologize if I am not following protocol correctly.Please let me know if so.)
(a b) (-b a)
which has non-zero determinant a2+b2 . We can induct, to show something similar holds for higher dimensions, i.e., the Jacobian ( of overlapping charts will necessarily be positive. Now, a couple of questions, please:
1)Is there a more topological proof of the orientability.?.I thought of using Lie theory, that Gl(n;C) is connected, may work, but I don't see how to rigorize this argument; in Milnor and Stasheff's book , it is stated that(paraphrase) this path connectedness allows one basis to be deformed into another homotopically, so that orientation is preserved. But, AFAIK, the bases are just elements in Cn. Any ideas on this direction.?. I know that in Cn, connectedness implies path-connectedness. I also know that Gl(n;C) can be embedded in Gl+(2n;IR), one of the connected components of Gl(2n;IR). I think this helps, but I don't know how to "rigorize" this idea. Any suggestions, please.?
I also wonder if one can generalize the CW-decomposition of CPn, where we can see that there is only one cell in the top dimension, to other complex manifolds.
2)Any suggestions, please, for showing that complex submanifolds S1,S2( of the "right dimensions, to make sure they can intersect) of a manifold M ,have positive intersection (self- or otherwise).? . I understand that, at every point p of intersection, we append the tangent spaces, defining:
and then the intersection is positive if there is an even permutation from the basis of TpM to the basis of Tp(S1/\S2). But I have no idea of how to show that, for complex submanifolds, the intersection is always positive.
Thanks For Any Help/Suggestions.