Existence of Embeddings of Smooth Varieties into Linear Algebraic Groups

TL;DR

This paper proves that every smooth affine variety can be embedded into sufficiently large simple algebraic groups, using advanced techniques involving principal bundles, transversality, and Chow groups, and discusses the optimality of the embedding bounds.

Contribution

It establishes the existence of embeddings of smooth affine varieties into simple algebraic groups with near-optimal dimension bounds, extending previous methods with new algebraic and topological insights.

Findings

Embeddings exist for dimensions at least 2d+2

The bound is nearly optimal, possibly improved to 2d+1

Homology calculations reveal limits of the embedding method

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.