Nontrivial vector bundles with trivial Chern classes

TL;DR

This paper constructs specific smooth affine algebras and projective modules over them with trivial total Chern class but nontrivial K-theory class, revealing subtle properties of vector bundles.

Contribution

It introduces new examples of vector bundles with trivial Chern classes that are nonetheless nontrivial in K-theory, expanding understanding of vector bundle classification.

Findings

Existence of smooth affine algebras with dimension p+2 for prime p

Construction of projective modules with trivial total Chern class

Identification of nontrivial K-theory classes despite trivial Chern classes

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 $C^r(P)=0 \Longrightarrow P\cong Q\oplus B$.