TITLE: A complex manifold which is quasiprojective in two different ways
QUESTION [13 upvotes]: 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...

REPLY [3 votes]: This may be relevant, although I'm not sure if it answers your question directly. The Russell cubic x+x^2y+t^2+z^3=0 is diffeomorphic to affine space A^3. But, it is known not to be algebraically isomorphic to A^3. I'm afraid I don't know the references for this.