Volume 18, issue 3 (2014)

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$K$–theory, LQEL manifolds and Severi varieties

Oliver Nash

Geometry & Topology 18 (2014) 1245–1260

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.

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$K$–theory, secant variety, Severi variety, quadric, dual variety
Mathematical Subject Classification 2010
Primary: 14M22
Secondary: 19L64
Received: 27 August 2013
Revised: 10 November 2013
Accepted: 12 December 2013
Published: 7 July 2014
Proposed: Lothar Goettsche
Seconded: Richard Thomas, Yasha Eliashberg
Oliver Nash