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Stiefel–Whitney numbers for singular varieties

Published online by Cambridge University Press:  18 January 2011

CARL McTAGUE
CARL McTAGUE*
Affiliation:
DPMMS, Wilberforce Road, Cambridge CB3 0WB. e-mail: c.mctague@dpmms.cam.ac.uk
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Abstract

This paper determines which Stiefel–Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F2-vector space of Stiefel–Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP3 bundles. These Stiefel–Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of 3 bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 273 - 289
Copyright
Copyright © Cambridge Philosophical Society 2011

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