Divisors defined by noncritical functions

TL;DR

This paper proves that in certain Stein manifolds, every complex hypersurface can be realized as the divisor of a holomorphic function with critical points exactly at the singularities, extending to higher codimension cases.

Contribution

It establishes a new link between complex hypersurfaces and holomorphic functions with controlled critical points in Stein manifolds.

Findings

Every complex hypersurface in a Stein manifold with trivial second cohomology is a divisor of a holomorphic function with critical points at singularities.

The result extends to complete intersections of higher codimension.

Provides a method to construct such holomorphic functions explicitly.

Abstract

In this paper we show that every complex hypersurface $A$ in a Stein manifold $X$ with $H^2(X;\mathbb Z)=0$ is the divisor of a holomorphic function $f$ on $X$ whose critical points are precisely the singular points of $A$. A similar result is proved for complete intersections of higher codimension.