Almost closed 1-forms

TL;DR

This paper constructs an algebraic almost closed 1-form with a zero scheme that cannot be locally expressed as a critical locus, addressing a question by Behrend-Fantechi and correcting previous work.

Contribution

It introduces a new algebraic example of an almost closed 1-form with a non-critically representable zero scheme, resolving an open question.

Findings

Zero scheme not expressible as a critical locus

Corrects previous incorrect construction

Answers Behrend-Fantechi's question

Abstract

We construct an algebraic almost closed 1-form with zero scheme not expressible (even locally) as the critical locus of a holomorphic function on a nonsingular variety. The result answers a question of Behrend-Fantechi. We correct here an error in our paper [MPT] where an incorrect construction with the same claimed properties was proposed.