Tame Discrete Sets on Affine Semisimple Homogeneous Spaces
Abstract
Winkelmann conjectures that every smooth flexible complex affine variety of dimension at least two is a Rosay--Rudin space. We verify this for every positive-dimensional quotient G/H with G a connected complex semisimple algebraic group and H a closed connected reductive subgroup. Weak and strong tameness coincide, every injection between tame discrete sets extends to a holomorphic automorphism, and complements of tame, finite, or empty sets are Oka. Every sufficiently sparse enumerated sequence can be sent to a fixed sequence by a composition of n complete holomorphic flow maps, with n≤6; injective self-maps of the fixed sequence admit such a realization with n≤2. These flows preserve an invariant algebraic volume form. The proof uses entire interpolation and regular functions invariant under two solvable subgroups to construct the automorphisms. If G is simple of rank at least two and G/H is spherical and admits a closed equivariant embedding into an irreducible module, the normalization bound improves to n≤4. We also prove that SL2/N(T), where T is a maximal torus, is an RR-space. This quotient is the complement of a smooth conic in P2.
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BibTeX
@misc{dvorsky2026tame,
title = {{Tame Discrete Sets on Affine Semisimple Homogeneous Spaces}},
author = {Alexander Dvorsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05479},
eprint = {2610.05479},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05479},
}
RIS
TY - UNPB
TI - Tame Discrete Sets on Affine Semisimple Homogeneous Spaces
AU - Alexander Dvorsky
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05479
UR - https://arxiv.org/abs/2610.05479
AB - Winkelmann conjectures that every smooth flexible complex affine variety of dimension at least two is a Rosay--Rudin space. We verify this for every positive-dimensional quotient G/H with G a connected complex semisimple algebraic group and H a closed connected reductive subgroup. Weak and strong tameness coincide, every injection between tame discrete sets extends to a holomorphic automorphism, and complements of tame, finite, or empty sets are Oka. Every sufficiently sparse enumerated sequence can be sent to a fixed sequence by a composition of n complete holomorphic flow maps, with n≤6; injective self-maps of the fixed sequence admit such a realization with n≤2. These flows preserve an invariant algebraic volume form. The proof uses entire interpolation and regular functions invariant under two solvable subgroups to construct the automorphisms. If G is simple of rank at least two and G/H is spherical and admits a closed equivariant embedding into an irreducible module, the normalization bound improves to n≤4. We also prove that SL_2/N(T), where T is a maximal torus, is an RR-space. This quotient is the complement of a smooth conic in P^2.
ER -