Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes

Abstract

We study smooth morphisms $f \colon X \to S$ that are $\mathbb{A}^1$-contractible in the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}(S)$. For base schemes $S$ of finite Krull dimension, we show that $\mathbb{A}^1$-contractibility is a fiberwise property: such a morphism is $\mathbb{A}^1$-contractible if and only if all its geometric fibers are $\mathbb{A}^1$-contractible. We apply this criterion to $\mathbb{A}^n$-fiber spaces, obtaining a geometric description of their $\mathbb{A}^1$-contractibility in terms of local factorizations as towers of torsors under vector bundles, building on results of Asanuma. In low relative dimensions, we establish rigidity results. In relative dimension $1$, $\mathbb{A}^1$-contractible morphisms over normal bases are precisely Zariski locally trivial $\mathbb{A}^1$-bundles. In relative dimension $2$, we show that over bases with characteristic zero residue fields, $\mathbb{A}^1$-contractible morphisms are $\mathbb{A}^2$-fiber spaces, and we obtain Zariski local triviality under additional hypotheses on the base. We also exhibit counterexamples in positive and mixed characteristic and formulate open problems concerning the existence of exotic $\mathbb{A}^1$-contractible surfaces.