Scheme-theoretic coisotropic reduction

ABSTRACT We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\mathbb {k}=\mathbb {R}$ or $\mathbb {k}=\mathbb {C}$, and is formulated for an affine symplectic groupoid $\mathcal {G}\,\,\substack{\longrightarrow \\ \longrightarrow }\,\,X$, an affine Hamiltonian $\mathcal {G}$-scheme $\mu :M\longrightarrow X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal {H}\,\,\substack{\longrightarrow \\\longrightarrow }\,\,S$. Our first main result is that the Poisson bracket on $\mathbb {k}[M]$ induces a Poisson bracket on the subquotient $\mathbb {k}[\mu ^{-1}(S)]^{\mathcal {H}}$. The Poisson scheme $M\mathbin {//}_{{S,\mathcal {H}}}\mathcal {G}:= \mathrm{Spec}(\mathbb {k}[\mu ^{-1}(S)]^{\mathcal {H}})$ is then declared to be a Hamiltonian reduction of M. Other main results include sufficient conditions for $M\mathbin {//}_{{S,\mathcal {H}}}\mathcal {G}$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to [6], where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden–Ratiu reduction [19], Mikami–Weinstein reduction [20], Śniatycki–Weinstein reduction [23], and symplectic reduction along general coisotropic submanifolds [6]. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore–Tachikawa conjecture.