On subelliptic manifolds

TL;DR

This paper proves that the blowup of a locally stably flexible manifold at a smooth submanifold is subelliptic and Oka, expanding understanding of flexibility and ellipticity in complex algebraic geometry.

Contribution

It establishes that blowups of locally stably flexible manifolds are subelliptic and Oka, linking flexibility properties with elliptic conditions in complex manifolds.

Findings

Blowup of a locally stably flexible manifold at a smooth submanifold is subelliptic.

Such blowups are also Oka manifolds.

Results are derived from properties of $k$-flexible manifolds.

Abstract

A smooth complex quasi-affine algebraic variety $Y$ is flexible if its special group $\SAut (Y)$ of automorphisms (generated by the elements of one-dimensional unipotent subgroups of $\Aut (Y)$) acts transitively on $Y$. An irreducible algebraic manifold $X$ is locally stably flexible if it is the union $\bigcup X_i$ of a finite number of Zariski open sets, each $X_i$ being quasi-affine, so that there is a positive integer $N$ for which $X_i\times \mathbb{C}^N$ is flexible for every $i$. The main result of this paper is that the blowup of a locally stably flexible manifold at a smooth algebraic submanifold (not necessarily equi-dimensional or connected) is subelliptic, and hence Oka. This result is proven as a corollary of some general results concerning the so-called $k$-flexible manifolds.