kirill@mathematik.uni-muenchen.de

Submission: 2007, Aug 16

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 similar fact for all $F_4$-varieties.

2000 Mathematics Subject Classification: 14C15; 14M15

Keywords and Phrases: splitting variety, Rost degree formula, algebraic cobordism

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