Local structure of the Teichm\"uller and the Riemann moduli stacks

The goal of this note is to introduce an interesting question proposed by D. Rydh on an analytic version of the local structure of Artin stacks saying that near points with linearly reductive stabilizers, Artin stacks are \'etale-locally quotient stacks. We give some supporting evidence by verifying it on two fundamental classes of classical analytic moduli spaces: the Teichm\"uller moduli space and the Riemann moduli space of integrable complex structures whose analytic stack versions have been constructed by a recent work of L. Meersseman.