Poisson $C^{\infty }$-schemes

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.

24 pages, v2: reorganization and some typos corrected, comments welcome. arXiv admin note: text overlap with arXiv:2503.19064