K-theory, LQEL manifolds and Severi varieties

TL;DR

This paper employs topological K-theory to analyze special algebraic varieties with quadratic entry loci, providing new proofs of classical theorems and resolving longstanding conjectures in algebraic geometry.

Contribution

It offers a novel K-theoretic approach to prove key properties of Severi varieties and related manifolds, including Zak's theorem and the Landman parity theorem.

Findings

Proof of Zak's theorem on Severi varieties dimensions

Resolution of Atiyah and Berndt's conjecture on Severi varieties

Application of K-theory to dual varieties and parity theorems

Abstract

We use topological K-theory to study non-singular 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 resolve a conjecture of Atiyah and Berndt. We also show how the same methods applied to dual varieties recover the Landman parity theorem.