Poisson $C^\infty$-schemes

TL;DR

This paper introduces Poisson $C^ty$-rings and schemes, establishing their properties and applications in singular symplectic and Poisson reductions, expanding the algebraic framework for Poisson geometry.

Contribution

It defines Poisson $C^ty$-rings and schemes, and explores their spectrum, providing a new algebraic approach to singular Poisson and symplectic reductions.

Findings

Spectrum of a Poisson $C^ty$-ring forms an affine Poisson $C^ty$-scheme

Applications include singular symplectic and Poisson reductions

Framework extends algebraic methods in Poisson geometry

Abstract

We introduce Poisson $C^\infty$-rings and Poisson local $C^\infty$-ringed spaces. We show that the spectrum of a Poisson $C^\infty$-ring is an affine Poisson $C^\infty$-scheme. We then discuss applications that include singular symplectic and Poisson reductions, singular quasi-Poisson reduction, coisotropic reduction and Poisson-Dirac subschemes.