The talk is based upon a joint work with Y. OHYAMA and J. SAULOY. Classically the space of Monodromy data (or character variety) of PV I (the sixth Painlevé diﬀerential equation) is the space of linear representations of the fundamental group of a 4-punctured sphere up to equivalence of representations. If one ﬁxes the local representation data it “is” a cubic surface. We will describe a $q$-analog: the space of $q$-Monodromy data of the $q$-Painlevé V I equation. For the $q$-analogs of the Painlevé equations (which are non-linear $q$-diﬀerence equations), according to H. SAKAI work, “everything” is well known on the “left side” of the ($q$-analog of the) Riemann-Hilbert map (the varieties of “initial conditions”), but the “right side” (the $q$-analogs of the spaces of Monodromy data or character varieties) remained quite mysterious.
We will present a complete description of the space of Monodromy data of $q$−PV I (some local data being ﬁxed). It is a “modiﬁcation” of an elliptic surface and we will explicit some “natural” parametrizations. This surface is analytically, but not algebraically isomorphic to the Sakai surface of ”initial conditions”. Our description uses a new tool, the Mano decompositions, which are a $q$-analog of the classical pants decompositions of surfaces. We conjecture that our constructions can be extended to the others $q$-Painlevé equations. This involves $q$-Stokes phenomena.