The associated map of the nonabelian Gauss-Manin connection
In ordinary Hodge theory for a compact kahler manifold, one can look at the Gauss-Manin connection as the complex structure of the manifold varies. The connection satisfies the Griffiths transversality property, and induces a map on the associated graded spaces of the Hodge filtered cohomology spaces of the manifolds. Similarly in nonabelian Hodge theory, where the manifolds are curves and nonabelian cohomology spaces are the moduli spaces of local systems on the curves, the Gauss-Manin connection will be the Isomonodromy deformation, which is a previously known structure on these moduli spaces. One can still define the Hodge filtration and calculate the map induced by the Gauss-Manin connection on the associated graded space. To do this we used deformation theory to express the tangent spaces of the moduli spaces as hypercohomologies of complexes of sheaves over the curves, and write the isomonodromy deformation as a map between such hypercohomology spaces. Under this setting the induced map can be explicitly calculated and is in fact written in an analogous form as the isomonodromy deformation. The induced map turns out to be closely related to another well-known structure called the Hitchin integrable structure, defined on the moduli spaces that correspond to the associated graded spaces. More specifically it is equal up to a factor of 2 to the quadratic Hitchin map.