Does there exist a complex manifold M which is a quasiprojective variety in two "essentially" different ways? Let me be more specific. I'm looking for a complex manifold M together with two projective varieties X and Y such that M is a locally closed Zariski dense subset of both X and Y. This gives two different notions of "algebraic" objects on M, and I want them to be different. For instance, can the resulting sheaves of regular function on M be different? Or the Picard groups? etc...
consider the punctured complex line C* and let M=C* x C*, seen as a holomorphic manifold.
You can embed M into P^1 X P^1 and get the standard algebraic affine structure on M, call it A.
On the other hand hand given an elliptic curve E there exists a P^1-bundle X over E such that, if you delete a section of this bundle X -->E, you get a Zariski open subset U of X which is algebraic, NOT AFFINE, but nevertheless biholomorphically isomorphic to M.
The Picard group of this U is isomorphic to that of E and so definitely not isomorphic to that of the affine variety A.
This construction is due to Serre and is mentioned in Hartshorne's "Algebraic Geometry" (Appendix B, Example 2.0.1, page 440).