Nontrivial vector bundles with trivial Chern classes
Satya Mandal
TL;DR
The paper addresses constructing nontrivial vector bundles with trivial Chern classes over smooth affine algebras in characteristic zero. It adapts Mohan Kumar's construction and the ABH splitting theorem to produce a smooth affine algebra $B$ of dimension $p+2$ and a projective $B$-module $Q$ of rank $p$ with $[Q]-[B^p] eq 0$ in $K_0(B)$ while the total Chern class $C(Q)$ is trivial. A key step is lifting from a MK framework to obtain a decomposition $Q\, ext{≅}\, Q_0oxplus B$ with $ ext{rank}(Q_0)=p$ and $C^k(Q_0)=0$ for all $1 o ext{dim }B$, thereby producing a nontrivial $K_0(B)$ class represented by a bundle with vanishing Chern classes. This work clarifies the relationship between $K_0$ and Chern data in higher dimensions and enriches the landscape of stably non-free, trivial-Chern-class modules.
Abstract
Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct projective $B$-modules $Q$ with $rank(Q)=p$, such that $x=[Q] -[B^p]\neq 0$ in $K_0(B)$ and the total Chern class $C(Q)=1+\sum_{i=1}^{p}C^k(Q) =1$ is trivial. We use the splitting theorem in \cite{ABH} that for projective $B$-modules $P$ with $rank(P)=r=\dim B-1$, vanishing $C^r(P)=0 \Longrightarrow P\cong Q\oplus B$.