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Utsav Choudhury

Publications and source records attributed to Utsav Choudhury.

10 recordsLinked to original sources
arXiv2026

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

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.

math.AG↗
arXiv2025

A Motivic Riemann-Roch Theorem for Deligne-Mumford Stacks

We develop a motivic cohomology theory, representable in the Voevodsky's triangulated category of motives, for smooth separated Deligne-Mumford stacks and show that the resulting higher Chow groups are canonically isomorphic to the higher $K$-theory of such stacks. This generalises the Grothendieck-Riemann-Roch theorem to the category of smooth Deligne-Mumford stacks.

math.AG↗
arXiv2024

$\mathbb{A}^1$-homotopy type of $\mathbb{A}^2 \setminus \left\{(0,0) \right\}$

In this article we prove that any $\mathbb{A}^1$-connected smooth $k$-variety is $\mathbb{A}^1$-uniruled for any algebraically closed field $k$. We establish that if a non empty open subscheme $X$ of a smooth affine $k$-scheme is $\mathbb{A}^1$-weakly equivalent to $\mathbb{A}^2_{k} \setminus \left\{(0,0) \right\}$, then $X \cong \mathbb{A}^2_{k} \setminus \left\{(0,0) \right\}$ as $k$-varieties for any field $k$ of characteristic $0$.

math.AG↗
arXiv2022

The Nisnevich Motive of an Algebraic Stack

We construct the motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes. We also show that for quotient stacks the motivic cohomology agrees with the Edidin-Graham-Totaro Chow groups with integer coefficients.

math.AG↗
arXiv2021

The Hurewicz map in motivic homotopy theory

For an $\A^1$-connected pointed simplicial sheaf $\sX$ over a perfect field $k$, we prove that the Hurewicz map $π_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$ is surjective. We also observe that the Hurewicz map for $¶^1_k$ is the abelianisation map. In the course of proving this result, we also show that for any morphism $ϕ$ of strongly $\A^1$-invariant sheaves of groups, the image and kernel of $ϕ$ are also strongly $\A^1$-invariant.

math.AG↗
arXiv2017

An isomorphism of motivic Galois groups

In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.

math.AG↗
arXiv2015

Homotopy theory of dg sheaves

In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.

math.CT↗
arXiv2012

On a conjecture of Morel

In this note we prove that the $\mathbb{A}^1$-connected component sheaf $a_{Nis}(π_0^{\mathbb{A}^1}(\mathcal{X}))$ of an $H$-group $\mathcal{X}$ is $\mathbb{A}^1$-invariant.

math.AG↗
arXiv2012

Motives of Deligne-Mumford Stacks

For every smooth and separated Deligne-Mumford stack $F$, we associate a motive $M(F)$ in Voevodsky's category of mixed motives with rational coefficients $\mathbf{DM}^{\eff}(k,\mathbb{Q})$. When $F$ is proper over a field of characteristic 0, we compare $M(F)$ with the Chow motive associated to $F$ by Toen (\cite{t}). Without the properness condition we show that $M(F)$ is a direct summand of the motive of a smooth quasi-projective variety.

math.AG↗