Blowup of a complex manifold at a point
= 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.