 Blowup of a complex manifold at a point

The blowup replaces a point of a complex $n$-manifold by the projective space of complex tangent directions ${\mathbb{CP}}^{n-1}$. A biholomorphism carrying one center to another lifts to a biholomorphism of their blowups.

It is the point-center case of blowup of an algebraic variety, also interpreted analytically.