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Journal of the London Mathematical Society Advance Access originally published online on September 29, 2007
Journal of the London Mathematical Society 2007 76(2):313-330; doi:10.1112/jlms/jdm061
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© 2007 London Mathematical Society

A formula for the Chern classes of symplectic blow-ups

Hansjörg Geiges

Mathematisches Institut
Universität zu Köln
Weyertal 86–90
50931 Köln
Germany

Federica Pasquotto

Afdeling Wiskunde
Vrije Universiteit
De Boelelaan 1081a
1081 HV Amsterdam
The Netherlands
pasquott@few.vu.nl

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.

2000 Mathematics Subject Classification 53D35, 57R20 (primary), 55N45, 32Q55 (secondary).

H. Geiges is partially supported by grant no. GE 1245/1-1 of the Deutsche Forschungsgemeinschaft (DFG) within the framework of the Schwerpunktprogramm 1154 ‘Globale Differentialgeometrie’. F. Pasquotto is supported by VIDI grant no. 639.032.202 of the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).

Received February 15, 2006; published online October 29, 2007.


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