Tame Discrete Sets on Affine Semisimple Homogeneous Spaces

Source 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 $n\le6$; injective self-maps of the fixed sequence admit such a realization with $n\le2$. 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 $n\le4$. 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 $\mathbf P^2$.