In this article we prove that every smooth affine variety of dimension d embeds into every simple algebraic group of dimension at least 2d + 2. For this we employ and build upon parametric transversality results for flexible affine varieties established by Kaliman yielding a certain embedding method. We show that our result is optimal up to a possible improvement by one to 2d + 1 by adapting a Chow-group based argument due to Bloch, Murthy, and Szpiro.In order to study the limits of our embedding method, we also show that there do not exist proper dominant maps from the d-dimensional affine space to any d-dimensional homogeneous space of a simple algebraic group. The latter is proved using a version of Hopf's theorem on the Umkehrungshomomorphismus from algebraic topology and rational homology group calculations. Contents 1. Introduction 1 2. Embeddings into the total space of a principal bundle 6 3. Applications: Embeddings into algebraic groups 16 4. Non-embedability results for algebraic groups 23 5. Limits of our methods for odd dimensional simple groups 24 Appendix A. Hopf's Umkehrungshomomorphismus theorem 28 Appendix B. A characterization of embeddings 33 Appendix C. The proof of Theorem C 35 References 38
We study the possible dynamical degrees of automorphisms of the affine space
$\mathbb {A}^n$
. In dimension
$n=3$
, we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space
$\mathbb {A}^n$
for some n, and we give the best possible n for quadratic integers, which is either
$3$
or
$4$
.
In this note we study the problem of characterizing the complex affine space A n via its automorphism group. We prove the following. Let X be an irreducible quasi-projective n-dimensional variety such that Aut(X) and Aut(A n ) are isomorphic as abstract groups. If X is either quasi-affine and toric or X is smooth with Euler characteristic χ(X) = 0 and finite Picard group Pic(X), then X is isomorphic to A n .The main ingredient is the following result. Let X be a smooth irreducible quasi-projective variety of dimension n with finite Pic(X). If X admits a faithful (Z/pZ) n -action for a prime p and χ(X) is not divisible by p, then the identity component of the centralizer Cent Aut(X) ((Z/pZ) n ) is a torus.
Let Y be the underlying variety of a connected affine algebraic group. We prove that two embeddings of the affine line C into Y are the same up to an automorphism of Y provided that Y is not isomorphic to a product of a torus (C * ) k and one of the three varieties C 3 , SL2, and PSL2.
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