The Liouville phenomenon in the deformation problem of coisotropics

TL;DR

This paper explores the deformation theory of coisotropic submanifolds using L-infinity algebras, linking foliation cohomology and leaf space models to understand obstructions and the Liouville phenomenon.

Contribution

It introduces a novel application of L-infinity algebra structures to coisotropic deformation problems and emphasizes the role of Haefliger's group in obstruction analysis.

Findings

L-infinity algebra effectively describes coisotropic deformations.

Obstruction equations' solvability relates to foliation cohomology.

Liouville/Diophantine distinction in KAM theory detected via L-infinity structures.

Abstract

The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (\Omega^*[1](\fol), d_\fol), which allows a concise description of deformations in terms of a Maurer-Cartan equation. Infinitesimal deformations are given by d_\fol-closed forms, and the relation between infinitesimal deformations and full deformations can be studied in terms of obstruction classes lying in the foliation cohomology H^*_\fol. Closely related to the foliation cohomology is Haefliger's group \Omega^*_c(T/H), an under-appreciated model for the leaf space of a foliation. We make integral use of this group in showing solvability and unsolvability of the obstruction equations. We also show the L-infinity apparatus to be capable of detecting the Liouville/diophantine distinction of KAM theory, and argue for the greater significance of Haefliger's integration-over-leaves map in passing this fine structure to a geometric model for the leaf space.