A formula for the Chern classes of symplectic blow‐ups

Abstract 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 singular cohomology ring) for blow‐ups of symplectic and complex manifolds. This was used by the second 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.