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.

On the blow-up formula of twisted de Rham cohomology

“…It is of importance to notice that many important invariants of complex manifolds hold such a similar formula, for example, the Deligne cohomology [2], the Chern classes [6] and the Dolbeault cohomology of holomorphic vector bundles [10,11], etc.…”

Lectures on controlled Reeb dynamics

“…There are a number of subtleties here, such as the question of uniqueness of the symplectic form on the blow-up, see [39, Section 7.1]. Nonetheless, one can make sense of the Chern classes of the blown-up symplectic manifold, see [17].…”

Chern classes of blow-ups

“…In [9], Hansjörg Geiges and Federica Pasquotto extend the classic blow-up formula of Section 1·1 to the case of symplectic, complex, and real manifolds; their method follows closely the proof of Lascu and Scott in [12], whose algebro-geometric ingredients they transfer to the topological environment.…”