The first, second and fourth Painlev\'{e} equations on weighted projective spaces

Abstract

The first, second and fourth Painlev\'{e} equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of the equations, a simple proof of the Painlev\'{e} property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of $\C P^3(p,q,r,s)$. In particular, for the first Painlev\'{e} equation, a well known Painlev\'{e}'s transformation is geometrically derived, which proves to be the Darboux coordinates of a certain algebraic surface with a holomorphic symplectic form. The affine Weyl group, Dynkin diagram and the Boutroux coordinates are also studied from a view point of the weighted projective space.