Special correspondences and Chow traces of Landweber-Novikov operations

TL;DR

This paper demonstrates that varieties with special correspondences preserve the rationality of low-codimension cycles, extending Vishik's results to all F4-varieties using algebraic cobordism and Landweber-Novikov operations.

Contribution

It generalizes Vishik's theorem from quadrics to all F4-varieties by employing algebraic cobordism and Chow trace techniques involving Landweber-Novikov operations.

Findings

Proves rationality preservation of cycles in varieties with special correspondences.

Extends Vishik's theorem to all F4-varieties.

Uses algebraic cobordism and Landweber-Novikov operations as main tools.

Abstract

We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves the rationality of cycles of small codimensions. This fact was proven by Vishik in the case of quadrics and played the crucial role in his construction of fields with $u$-invariant $2^r+1$. The main technical tools are algebraic cobordism of Levine-Morel, generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. As a direct application of our methods we prove the Vishik's Theorem for all $F_4$-varieties.