K–theory, LQEL manifolds and Severi varieties

Nash, Oliver

doi:10.2140/gt.2014.18.1245

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$K$–theory, LQEL manifolds and Severi varieties

Oliver Nash

Geometry & Topology 18 (2014) 1245–1260

DOI: 10.2140/gt.2014.18.1245

Abstract

We use topological $K$ –theory to study nonsingular varieties with quadratic entry locus. We thus obtain a new proof of Russo’s divisibility property for locally quadratic entry locus manifolds. In particular we obtain a $K$ –theoretic proof of Zak’s theorem that the dimension of a Severi variety must be $2$ , $4$ , $8$ or $16$ and so answer a question of Atiyah and Berndt. We also show how the same methods applied to dual varieties recover the Landman parity theorem.

Keywords

$K$–theory, secant variety, Severi variety, quadric, dual variety

Mathematical Subject Classification 2010

Primary: 14M22

Secondary: 19L64

References

Publication

Received: 27 August 2013

Revised: 10 November 2013

Accepted: 12 December 2013

Published: 7 July 2014

Proposed: Lothar Goettsche

Seconded: Richard Thomas, Yasha Eliashberg

Authors

Oliver Nash
Dublin Ireland
http://www.olivernash.org

Volume 18, issue 3 (2014)