Curvature of Poisson pencils in dimension three

TL;DR

This paper introduces a new explicit curvature form to determine the flatness of Poisson pencils on 3-manifolds, simplifying the analysis and revealing many non-flat examples among linear pencils on Lie algebras.

Contribution

It proposes a simpler obstruction called the curvature form, providing an explicit formula and applying it to classify flatness of linear pencils on Lie algebras.

Findings

Curvature form can be explicitly computed for Poisson pencils.

Many linear pencils on Lie algebras are non-flat.

The curvature form simplifies flatness criteria for Poisson pencils.

Abstract

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil. We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.