Existence of Embeddings of Smooth Varieties into Linear Algebraic Groups

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.

arXiv comment54 pages, comments are welcome! In V3 we fixed an error in the proof of Proposition 2.15. Furthermore, we provide a more general version of Theorem C so that it recovers Gurjar's Theorem. This version is accepted for publication in the Journal of Algebraic Geometry