Existence of embeddings of smooth varieties into linear algebraic groups

Feller, Peter; van Santen, Immanuel

doi:10.1090/jag/793

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Existence of embeddings of smooth varieties into linear algebraic groups

Authors: Peter Feller and Immanuel van Santen
Journal: J. Algebraic Geom. 32 (2023), 729-786
DOI: https://doi.org/10.1090/jag/793
Published electronically: October 3, 2022
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Abstract | References | Additional Information

Abstract:

We prove that every smooth affine variety of dimension $d$ embeds into every simple algebraic group of dimension at least $2d+2$. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to $2d+1$.

In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.

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Additional Information

Peter Feller
Affiliation: ETH Zürich, Department of Mathematics, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID: 1052130
Email: peter.feller@math.ch

Immanuel van Santen
Affiliation: University of Basel, Department of Mathematics and Computer Science, Spiegelgasse $1$, CH-$4051$ Basel, Switzerland
MR Author ID: 997932
ORCID: 0000-0002-9903-829X
Email: immanuel.van.santen@math.ch

Received by editor(s): May 24, 2021
Received by editor(s) in revised form: July 16, 2021, and August 9, 2021
Published electronically: October 3, 2022
Additional Notes: The first author was supported by the SNSF Grant 181199
Article copyright: © Copyright 2022 University Press, Inc.

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