Shifted Lagrangian thickenings of shifted Poisson derived schemes

TL;DR

This paper establishes an equivalence between shifted Poisson structures and shifted Lagrangian thickenings on derived schemes, and extends shifted symplectic results to shifted Poisson structures, with implications for mapping stacks.

Contribution

It proves a conjecture linking shifted Poisson structures to Lagrangian thickenings and extends shifted symplectic theorems to the Poisson setting.

Findings

Equivalence between shifted Poisson structures and Lagrangian thickenings.

Mapping stacks inherit shifted Poisson structures under certain conditions.

Extends known symplectic results to the Poisson context.

Abstract

We prove that the space of shifted Poisson structures on a derived scheme $X$ locally of finite presentation is equivalent to the space of shifted Lagrangian thickenings out $X$, solving a conjecture in shifted Poisson geometry. As a corollary, we show that for $M$ a compact oriented $d$-dimensional manifold and an $n$-shifted Poisson structure on $X$, the mapping stack $\mathrm{Map}(M,X)$ has an $(n-d)$-shifted Poisson structure. It extends a known theorem for shifted symplectic structures to shifted Poisson structures.