#### Volume 18, issue 3 (2014)

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$K$–theory, LQEL manifolds and Severi varieties

### Oliver Nash

Geometry & Topology 18 (2014) 1245–1260
##### 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
Primary: 14M22
Secondary: 19L64