Thom's jet transversality theorem for regular maps

TL;DR

This paper proves a version of Thom's jet transversality theorem for algebraic regular maps, extending classical results to algebraic and flexible manifolds, with applications to holomorphic submersions and Oka properties.

Contribution

It establishes an algebraic jet transversality theorem for regular maps from affine algebraic manifolds to flexible algebraic manifolds, generalizing holomorphic results.

Findings

Genericity theorems for regular maps of maximal ranks

Every connected compact locally flexible manifold is a holomorphic submersion image from affine space

Algebraically degenerate subvarieties of codimension ≥2 have Oka complements

Abstract

We establish Thom's jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstneri\v{c}'s jet transversality theorem for holomorphic maps from a Stein manifold to an Oka manifold. Our jet transversality theorem implies genericity theorems for regular maps of maximal ranks. As an application, it follows that every connected compact locally flexible manifold is the image of a holomorphic submersion from an affine space. We also show that every algebraically degenerate subvariety of codimension at least two in a locally flexible manifold has an Oka complement.