In the case of a line bundle over M, positivity of such a bundle (one whose curvature form which is Kahler) gives rise to an embeddings of M into the projective space.
Now I have in mind (more or less) the following definition. Let E be a holomorphic vector bundle over M with a hermitian metric. Moreover let D be a connection on E. Then we can define D^2 so that the curvature matrix of 2-forms. Such a curvature matrix (tensor) gives rise to a Hermitian form O_E on the bundle TM\otimes E. We can say that E is positive if such a hermitian form O_E is positive on all the tensors in TM\otimes E.
Then, What is the geometric meaning (if any) of the positivity in a vector bundle? rank>1.
I think, there are several definitions that generalize the concept of positive line bundle. Can you say which is the more standard one and why?
I edited the previous question since it was ambiguous.
Best Answer
I think there might be some confusion between the following notions:
A complex vector bundle on a manifold (yields a map to BGLn(C)).
A holomorphic vector bundle on a complex manifold (gives an embedding to projective space when "positive" in a suitable sense).
This distinction confused me for a while. A holomorphic structure on a bundle is not a trivial thing. It roughly amounts to half of the data of a connection, by forcing holomorphicity on local sections.