Algebraic subellipticity and dominability of blow-ups of affine spaces

TL;DR

This paper investigates how the Oka property and algebraic subellipticity behave under blow-ups of complex manifolds, proving that certain properties are preserved when blowing up along algebraic submanifolds.

Contribution

It establishes that algebraic subellipticity is preserved under blow-ups along arbitrary algebraic submanifolds, extending known results for points and complements of codimension two subvarieties.

Findings

Algebraic subellipticity is preserved under blow-ups along algebraic submanifolds.

Strong algebraic dominability is preserved under blow-ups with smooth centers.

Confirmed that every open Riemann surface can be properly embedded into C2.

Abstract

Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class $\mathscr A$ if it is the complement of an algebraic subvariety of codimension at least $2$ in an algebraic manifold that is Zariski-locally isomorphic to $\mathbb C^n$. A manifold of Class $\mathscr A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\mathscr A$ blown up at finitely many points is of Class $\mathscr A$. Our main result is that a manifold of Class $\mathscr A$ blown up along an arbitrary algebraic submanifold (not necessarily connected) is algebraically subelliptic. For algebraic manifolds in general, we prove that strong algebraic dominability, a weakening of algebraic subellipticity, is preserved by an arbitrary blow-up with a smooth centre. We use the main result to confirm a prediction of Forster's famous conjecture that every open Riemann surface may be properly holomorphically embedded into $\mathbb C^2$.