"Algebraic geometry is more rigid than differential geometry or topology" I am reading Robin Hartshorne's Algebraic Geometry. In chapter $1$, section $4$, he says:

Since an open subset of a variety is dense, this already carries a lot of information. In this respect algebraic geometry is more "rigid" than differential geometry or topology. In particular, the concept of birational equivalence is unique.

What does he mean? I did not understand what does he mean when he says that "algebraic geometry is more "rigid" than differential geometry or topology".

At the simplest level, one can say that differential geometry is more "flabby" because it admits things like partitions of unity, which are necessarily defined by nonanalytic functions.  An algebraic geometry the defining transition functions are by definition at least rational and therefore analytic.  An analytic function is completely determined by how it behaves on a dense set in a neighborhood of a point.
