A formula for the Chern classes of symplectic blow-ups

It is shown that the formula for the Chern classes (in the Chow ring) of blow-ups of algebraic varieties, due to Porteous and Lascu-Scott, also holds (in the cohomology ring) for blow-ups of symplectic and complex manifolds. This was used by the second-named author in her solution of the geography problem for 8-dimensional symplectic manifolds. The proof equally applies to real blow-ups of arbitrary manifolds and yields the corresponding blow-up formula for the Stiefel-Whitney classes. In the course of the argument the topological analogue of Grothendieck's `formule clef' in intersection theory is proved.