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

Utsav Choudhury; Aritra Mandal; Biman Roy

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arXiv · 2608.10966

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

Utsav Choudhury·Aritra Mandal·Biman Roy

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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.

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Utsav Choudhury, Aritra Mandal, Biman Roy. 2026-08-11. $\mathbb{A}^1$-fibration in algebraic geometry and $\mathbb{A}^1$-homotopy type. https://arxiv.org/abs/2608.10966

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