Topologically, a classIn the case of isomorphisma line bundle over M, positivity of such a complex vector bundle E-->X (of rank n)one whose curvature form which is equivalentKahler) 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 homotopy classholomorphic 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 its classifying map X-2->BGforms. Such a curvature matrix (GL_ntensor) gives rise to a Hermitian form O_E on the bundle TM\otimes E. Then 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? n>1rank>1. Intuitively
I think, can such a classifying map turn out to be an embedding for a specific type of bundle? as it is inthere are several definitions that generalize the caseconcept of positive line bundlesbundle. Can you say which is the more standard one and why?
I edited the previous question since it was ambiguous.