Closed symmetric 3-differentials on complex surfaces

TL;DR

This paper establishes a precise criterion for when a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface can be locally expressed as a product of three closed holomorphic 1-forms, addressing a question in complex geometry.

Contribution

It provides a necessary and sufficient condition for representing symmetric 3-differentials as products of closed holomorphic 1-forms, including a coordinate-free differential operator version.

Findings

Derived a condition for local representability as a product of closed forms

Introduced a coordinate-free differential operator for the condition

Answered a question posed by Bogomolov and de Oliveira

Abstract

We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give two versions of this condition corresponding to different choices of coordinates, one of which defines a coordinate-free differential operator, answering a question of Bogomolov and de Oliveira.