Almost Lagrangian Obstruction

TL;DR

This paper investigates the obstruction to transforming an almost Lagrangian fibration into a true Lagrangian one, linking it to cohomological cup products and providing explicit examples of non-trivial obstructions.

Contribution

It explicitly describes the Lagrangian obstruction via the Dazord-Delzant homomorphism and offers the first concrete examples of non-trivial obstructions.

Findings

Obstruction related to cup product in cohomology with local coefficients.

Explicit description of the Dazord-Delzant homomorphism.

First examples of non-trivial Lagrangian obstructions.

Abstract

The aim of this paper is to describe the obstruction for an almost Lagrangian fibration to be Lagrangian, a problem which is central to the classification of Lagrangian fibrations and, more generally, to understanding the obstructions to carry out surgery of integrable systems, an idea introduced by Zung. It is shown that this obstruction (namely, the homomorphism of Dazord and Delzant) is related to the cup product in cohomology with local coefficients on the base space B of the fibration. The map is described explicitly and some examples are calculated, thus providing the first examples of non-trivial Lagrangian obstructions.