Elliptic bihamiltonian structures from relative shifted Poisson structures

TL;DR

This paper constructs shifted Poisson structures on moduli stacks over Calabi-Yau fibrations, leading to new elliptic bihamiltonian structures and extending known Poisson brackets on projective spaces.

Contribution

It generalizes previous constructions to include singular fibers, producing new examples of compatible Poisson brackets related to elliptic bihamiltonian structures.

Findings

Constructed shifted Poisson structures on moduli stacks

Derived new elliptic bihamiltonian structures from these constructions

Explicitly recovered known brackets on Hirzebruch surfaces

Abstract

In this paper, generalizing the construction of \cite{HP1}, we equip the relative moduli stack of complexes over a Calabi-Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin-Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii-Wolf in \cite{OW}.