Symplectic resolutions of the quotient of R^2 by a non-finite symplectic group

Abstract

We construct smooth symplectic resolutions of the quotient of R^2 under some infinite discrete sub-group of GL_2(R) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val symplectic resolution of C^2/G, with G \subset SL_2(C) a finite group. The first of these infinite groups is G=Z, identified to triangular matrices with spectrum {1}. Smooth functions on the quotient R^2/G come with a natural Poisson bracket, and R^2/Gis for an arbitrary k \geq 1 set-isomorphic to the real Du Val singular variety A_{2k} = {(x,y,z) \in 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 R^2/G. The same holds for G'=Z \rtimes Z/2Z (identified to triangular matrices with spectrum {\pm 1}), with the upper half of D_{2k+1} playing the role of A_{2k}.