Isomonodromic deformation of Fuchsian projective connections on elliptic curves

We consider isomonodromic deformations of second-order Fuchsian differential equations on elliptic curves. Our primary object of study is the monodromy mapping F : epsilon --> R, where epsilon is the space of deformation parameters of Fuchsian equations, and R is the space of conjugacy classes of representations of the fundamental group of the nonsingular locus of the equations. Our goal is to describe the tangential directions to the fibers of F by a completely integrable system of partial differential equations on the space epsilon. For this purpose we use the fact that there is defined a canonical closed nondegenerate 2-form (i.e., a symplectic structure) w on the space R. By pulling back this 2-form onto epsilon via F, we obtain a closed 2-form F*w that describes the isomonodromic deformations as the directions making F*w degenerate. As a result we find that the isomonodromic deformations are described as a completely integrable Hamiltonian system. Our specific setting of Fuchsian equations on elliptic curves was originally treated by Okamoto [16]-[18] and then generalized by Iwasaki [6] to the case of higher genera. The main new features of our discussion here are that (i) we allow the underlying elliptic curve also vary, and that (ii) we use fully the "pulling-back" principle which was inspired by Iwasaki [7]. Isomonodromic deformations on elliptic curves have also appeared in Korotkin-Samtleben [13], Levin-Olshanetsky [14] and Takasaki [19], [20] via different methods and motivations. This article is organized as follows: In Section 1 we introduce the linear equations, illustrate our method of studying isomonodromic deformations, and then present our main results. Section 2 contains the construction of the space R of representations with particular emphasis on its smooth structure and the description of its tangent spaces. In Section 3 we develop the variational theory of monodromy representations. Section 4 contains the description of the symplectic structure on R. In view of the results of the preceding two sections, we evaluate the symplectic form on R in Section 5. The details of the calculation are given in Section 6 and the explicit form of the pulled-back 2-form is then obtained. Analyzing that 2-form closely, we derive the desired isomonodromy equation in Section 7. Finally in Section 8 we examine in more detail the isomonodromy equation in the simplest setting. This article is a revised and extended version of my thesis [9] at the University of Tokyo, 1995. Part of the results here has been announced in [11].