Complex structure on a vector space defines the orientation. Two components are complex structures with the opposite orientation. The correspondence $I \mapsto -I$ identifies these two components when dimension is odd, and produces an involution on the Teichmuller space of complex structures when it is even.

Many thanks! Is it possible to obtain an estimate of the Ricci flow if we start from a metric which is flat on an open ball?

Then it follows from the valuative criterion of properness, indeed

Projective subvarieties are compact, and compact things are closed. Though you did not specify the topology, which one do you want?

It is not clear what do you mean by $J$, but I guess it is the complex structure in the bundle. Then you are asking how to verify that the Chern connection is complex linear. There is no reason to verify this, it is complex linear by definition (its structure group is $U(n)$).

This is why I was talking about finite maps! The idea is to take a finite quotient, and then an embedding. Though the cyclic group cannot act by permutation of the factors, and I was asking about the cyclic group action, so I am not sure we cannot have an embedding.

sure, but on top of it we also have two algebraic structures with the same complex structure

thanks! Yes, it was so hard to find it, I managed and then lost the memory of finding it...