Holme type theorem for special linear groups

TL;DR

This paper establishes conditions under which certain algebraic varieties can be embedded into special linear groups or their products, expanding understanding of embeddings in algebraic geometry.

Contribution

It proves new embedding theorems for algebraic varieties into special linear groups and their products, based on dimension and flexibility conditions.

Findings

Embedding of Z into X when dim X > ED(Z) - 1

Embedding of Z into Y x affine n-space under specified conditions

Extension of embedding results to flexible varieties and product spaces

Abstract

Let Z be an affine algebraic variety and ED(Z)= max(2 dim Z+1, dim TZ). Let X be a smooth algebraic variety isomorphic to a semi-simple linear algebraic group whose Lie algebra is a sum of special linear Lie algebras. We show that if dim X > ED(Z) -1, then Z admits a closed embedding into X. We also show that for every smooth affine flexible variety Y there is a closed embedding of $Z$ into the the product of Y and an affine n-space provided that n > dim Z- 2 and dim Y +n > ED (Z) -1.