GENERALIZED MONODROMY DATA

ABSTRACT. In this note, we will give a brief summary of geometric approach to understanding equations of Painlev\’e type ( $[0] $ , [Sakai], [STT], [IISI], [In]). Finally, we report the recent result on the moduli space of the generalized monodromy data associated to 10 families of isomonodromic problem related to the classical Painlev\’e equations in [PSa]. 1. GEOMETRIC APPROACH To EQUATIONS OF PAINLEV\’E TYPES First of all, we would like to explain about differential equations of Painlev\’e type and known geometric approach to understand the equations. 1.1. Differential equations of Painlev\’e type. A complex algebraic ordinary differential equation is said to have the Painleve property, if its all solutions has no movable singularities other than poles. We call an algebraic ODE which satisfies Painlev\’e property an ODE of Painleve type. Moreover, we can naturally extend the definition of Painlev\’e property for a partial differential equation, so we will also use $t\}_{1e} $ terni ” an equation of Painleve type”. After a result due to L. Fuchs and H. Poincar\’e for the first order case, P. Painlev\’e and his former student B. Gambier classified the rational differential equation of order two $q”=R(t, q, q’) $ which may satisfy Painlev\’e property into 6 types, $P_{I}, $ $\cdots, $ $P_{VI} $. W