Poisson Cohomology of Broken Lefschetz Fibrations

TL;DR

This paper computes the Poisson cohomology of broken Lefschetz fibrations by analyzing singularities, extending techniques from Sklyanin algebra, and providing formulas for Jacobian Poisson structures in 4D.

Contribution

It introduces a method to compute Poisson cohomology at singularities of broken Lefschetz fibrations and offers new formulas for Jacobian Poisson structures in four dimensions.

Findings

Poisson cohomology at fold singularities reduces to 3D point singularity calculations.

Techniques from Sklyanin algebra are adapted for singularity analysis.

Provides compact formulas for Poisson coboundary operators in 4D Jacobian structures.

Abstract

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.