Algebraic cobordism and flag varieties

TL;DR

This paper investigates torsion elements in the Chow ring of algebraic varieties with cellular structure, focusing on their relation to algebraic cobordism, especially for twisted flag varieties, to enhance understanding of their algebraic and topological properties.

Contribution

It provides a method to compute the Chow ring from algebraic cobordism for twisted complete flag varieties, linking torsion elements to cobordism generators.

Findings

Identified torsion elements in the Chow ring related to cobordism generators.

Developed a computational approach for Chow rings of twisted flag varieties.

Enhanced understanding of the relationship between algebraic cobordism and Chow rings.

Abstract

Let $X$ be an algebraic variety over $k$ such that $\bar X=X\otimes_k\bar k$ is cellular. We study torsion elements in the Chow ring $CH^*(X)$ which corresponds to $v_iy$ in the algebraic cobordism $\Omega^*(\bar X)$ where $0\not=y\in CH^*(\bar X)/p$ and $v_i$ is the generator of $BP^*$ with $|v_i|=-2(p^i-1).$ In particular, we try to compute $CH^*(X)$ from $\Omega^*(\bar X)$ when $X$ are twisted complete flag varieties.