Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres
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2026
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| author | Vijayalakshmi, Krishna Kumar Madhavan |
| author_facet | Vijayalakshmi, Krishna Kumar Madhavan |
| contents | One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions $n<3$ over certain fields. In this thesis, we extend this observation to "reasonably" arbitrary base schemes in relative dimensions $d<3$, exploiting the Zariski local triviality and the triviality of the sheaf of relative differentials. From dimensions $n\geq 3$, the existence of smooth "exotic" affine schemes - those that are $\mathbb{A}^1$-contractible but not isomorphic to the affine $n$-space - has already been established. A well-studied family constitutes the Koras-Russell threefolds $\mathcal{K}$ and their higher-dimensional prototypes $\mathcal{X}_n$, whose $\mathbb{A}^1$-contractibility has been so far proven over fields of characteristic zero. Here, we extend the relative $\mathbb{A}^1$-contractibility of $\mathcal{K}$ and $\mathcal{X}_n$ over a Noetherian base scheme in arbitrary dimensions. Then, using these prototypes, we study the existence of "exotic spheres" - $n$-dimensional smooth schemes that are $\mathbb{A}^1$-homotopic, but not isomorphic to $\mathbb{A}^n \backslash \{0\}$ - in motivic homotopy theory. This result can be seen as the "compact" analog of the study of exotic affine schemes. Our main result shows that in all dimensions $n\geq 4$, the quasi-affine varieties $\mathcal{X}_n \backslash \{\bullet\}$ give a model for the exotic motivic spheres over infinite perfect fields. The novelty is that these constitute the first family of examples of smooth motivic spheres of dimension $n$, which are not isomorphic to $\mathbb{A}^n \backslash \{0\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_04167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres Vijayalakshmi, Krishna Kumar Madhavan Algebraic Geometry Commutative Algebra Algebraic Topology 14F42, 14R05, 14R10 (Primary), 14F45, 14R25, 55M99 (Secondary) One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions $n<3$ over certain fields. In this thesis, we extend this observation to "reasonably" arbitrary base schemes in relative dimensions $d<3$, exploiting the Zariski local triviality and the triviality of the sheaf of relative differentials. From dimensions $n\geq 3$, the existence of smooth "exotic" affine schemes - those that are $\mathbb{A}^1$-contractible but not isomorphic to the affine $n$-space - has already been established. A well-studied family constitutes the Koras-Russell threefolds $\mathcal{K}$ and their higher-dimensional prototypes $\mathcal{X}_n$, whose $\mathbb{A}^1$-contractibility has been so far proven over fields of characteristic zero. Here, we extend the relative $\mathbb{A}^1$-contractibility of $\mathcal{K}$ and $\mathcal{X}_n$ over a Noetherian base scheme in arbitrary dimensions. Then, using these prototypes, we study the existence of "exotic spheres" - $n$-dimensional smooth schemes that are $\mathbb{A}^1$-homotopic, but not isomorphic to $\mathbb{A}^n \backslash \{0\}$ - in motivic homotopy theory. This result can be seen as the "compact" analog of the study of exotic affine schemes. Our main result shows that in all dimensions $n\geq 4$, the quasi-affine varieties $\mathcal{X}_n \backslash \{\bullet\}$ give a model for the exotic motivic spheres over infinite perfect fields. The novelty is that these constitute the first family of examples of smooth motivic spheres of dimension $n$, which are not isomorphic to $\mathbb{A}^n \backslash \{0\}$. |
| title | Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres |
| topic | Algebraic Geometry Commutative Algebra Algebraic Topology 14F42, 14R05, 14R10 (Primary), 14F45, 14R25, 55M99 (Secondary) |
| url | https://arxiv.org/abs/2603.04167 |