Symplectic resolutions of the quotient of \( {{\mathbb {R}}}^2 \) by an infinite symplectic discrete group

Abstract

We construct smooth symplectic resolutions of the quotient of \({\mathbb {R}}^2 \) under some infinite discrete sub-group of \({\textrm{ GL}}_2({\mathbb {R}}) \) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val symplectic resolution of \({\mathbb {C}}^2 \hspace{-1.5pt} / \hspace{-1.5pt}G\), with \(G \subset {\textrm{ SL}}_2({\mathbb {C}}) \) a finite group. The first of these infinite groups is \(G={\mathbb {Z}}\), identified to triangular matrices with spectrum \(\{1\} \). Smooth functions on the quotient \(\mathbb {R}^2 \hspace{-1.5pt} / \hspace{-1.5pt} G \) come with a natural Poisson bracket, and \(\mathbb {R}^2\hspace{-1.5pt} / \hspace{-1.5pt}G\) is for an arbitrary \(k \ge 1\) set-isomorphic to the real Du Val singular variety \(A_{2k} = \{(x,y,z) \in {\mathbb {R}}^3, x^2 +y^2= z^{2k}\}\). We show that each one of the usual minimal resolutions of these Du Val varieties are symplectic resolutions of \(\mathbb {R}^2\hspace{-1.5pt} / \hspace{-1.5pt}G\). The same holds for \(G'={\mathbb {Z}} \rtimes {\mathbb {Z}}\hspace{-1.5pt} / \hspace{-1.5pt}2\mathbb {Z}\) (identified to triangular matrices with spectrum \(\{\pm 1\} \)), with the upper half of the Du Val singularity \(D_{2k+1} \) playing the role of \(A_{2k}\).