Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of $\bold{P}^2$

Abstract

An algebro-geometric setting for the study of the Painlevé VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two. Relations with the theory of elliptic functions and the quantum cohomology of projective plane are discussed.