$K$–theory, LQEL manifolds and Severi varieties

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.