Understanding why nonsingular complex algebraic varieties are analytic manifolds.

This follows from the complex-analytic version of the implicit function theorem. Nonsingularity of the variety is equivalent to nonvanishing of the Jacobian, which is precisely the condition required to apply the theorem.

The general result is that a smooth complex algebraic variety naturally yields a complex manifold. In more detail, Serre constructed the analytification functor from complex algebraic varieties to complex analytic spaces (see his paper GAGA). Essentially, this functor converts the Zariski topology of a variety into a complex-analytic topology. Also, it sends smooth varieties to complex manifolds.

This is the most rigorous and systematic approach to passing between algebraic and differential geometry of which I am aware.