arXiv subjects

Krishna Kumar Madhavan Vijayalakshmi

Publications and source records attributed to Krishna Kumar Madhavan Vijayalakshmi.

3 recordsLinked to original sources

arXiv2026

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

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.

math.AG↗

arXiv2026

Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres

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 $\mathbb{A}^n \backslash \{0\}$.

math.AG↗

arXiv2025

Relative $\mathbb{A}^1$-Contractibility of Koras-Russell Prototypes and Exotic Motivic Spheres

The Koras-Russell threefolds are a certain family of smooth, affine contractible threefolds exhibiting "exotic" behavior in the algebro-geometric context. Our goal in this note is to extend its $\mathbb{A}^1$-contractibility from a field to a general base scheme. As a consequence, we also give a general strategy to extend the $\mathbb{A}^1$-contractibility of Koras-Russell prototypes in higher dimensions over a general base scheme. As a major consequence, we establish the existence of "exotic" motivic spheres in all dimensions at least 4 over infinite perfect fields.

math.AG↗