$\mathbb{A}^1$-fibration in algebraic geometry and $\mathbb{A}^1$-homotopy type

Utsav Choudhury; Aritra Mandal; Biman Roy

$\mathbb{A}^1$-fibration in algebraic geometry and $\mathbb{A}^1$-homotopy type

Algebraic Geometry 2026-08-11 v1

Authors: Utsav Choudhury , Aritra Mandal , Biman Roy

Abstract

In this article we show that an $\mathbb{A}^1$ bundle map or a vector bundle map $p: X \to Y$ induces trivial local fibration $\underline{Sing}(X) \to \underline{Sing}(Y)$ . Using this, we first show that for Korus Russel threefolds of first kind $X$ the space $\underline{Sing}(X)$ is $\mathbb{A}^1$ local. Then we show that for any smooth affine complex surface $X$ , the $\mathbb{A}^1$ - connected component sheaf is homotopy invariant.

Keywords

$\mathbb{A}^1$-fibration $\mathbb{A}^1$-homotopy type algebraic geometry Korus-Russel threefolds homotopy invariance

Cite

@article{arxiv.2608.10966,
  title  = {$\mathbb{A}^1$-fibration in algebraic geometry and $\mathbb{A}^1$-homotopy type},
  author = {Utsav Choudhury and Aritra Mandal and Biman Roy},
  journal= {arXiv preprint arXiv:2608.10966},
  year   = {2026}
}