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On the numerical triviality of $BP$-cycles

Alexander Vishik

TL;DR

This work studies numerical triviality of $BP^*/I(\infty)^r$-classes by relating it to isotropy of associated varieties over the flexible closure $\widetilde{k}$. Using INCHKm, deformation to the normal cone, and symmetry operations, it shows that isotropy and numerically trivial information align for $BP^*/I$-type theories, and that for the prime $2$ this data is captured by a coherent chain of pure symbols in $K^M_*(\widetilde{k})/2$. It further demonstrates that $BP^*/I(\infty)^r$ acts as a regular substitute for $\operatorname{CH}^*/p^r$ in isotropy arguments, reducing BP-numerical triviality to CH$^*/p$ anisotropy and ultimately to CH$^*/p$ anisotropy via symmetric powers. The results yield explicit, symbol-based criteria for numerical triviality and establish a unified framework tying isotropy, numerical equivalence, and Chow-theory mod $p$ through geometric reductions.

Abstract

We show that, in the case of a prime $2$, the numerical triviality of $BP$-cycles modulo various powers of the (augmentation) invariant ideal $I(\infty)$ is controlled by pure symbols in $K^M_*/2$ over the flexible closure of the base field.

On the numerical triviality of $BP$-cycles

TL;DR

This work studies numerical triviality of -classes by relating it to isotropy of associated varieties over the flexible closure . Using INCHKm, deformation to the normal cone, and symmetry operations, it shows that isotropy and numerically trivial information align for -type theories, and that for the prime this data is captured by a coherent chain of pure symbols in . It further demonstrates that acts as a regular substitute for in isotropy arguments, reducing BP-numerical triviality to CH anisotropy and ultimately to CH anisotropy via symmetric powers. The results yield explicit, symbol-based criteria for numerical triviality and establish a unified framework tying isotropy, numerical equivalence, and Chow-theory mod through geometric reductions.

Abstract

We show that, in the case of a prime , the numerical triviality of -cycles modulo various powers of the (augmentation) invariant ideal is controlled by pure symbols in over the flexible closure of the base field.
Paper Structure (4 sections, 6 theorems, 3 equations)

This paper contains 4 sections, 6 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Isotropic and numerical equivalence
  3. $p$-primary Chow groups vis $I(\infty)$-primary $BP$-theory
  4. The case of prime $2$

Key Result

Theorem 2.4

Let $Q^*=BP^*/I$ be a free oriented cohomology theory, where $I\subset BP$ is a non-zero ideal invariant under Landweber-Novikov operations. Let $X$ be a smooth projective variety and $u\in Q^*(X)$ be a numerically trivial element. Then, over some purely transcendental extension $k({\Bbb P})$ of $k$

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Proposition 4.1
  • Theorem 4.2
  • ...and 1 more