Algebraic Geometry Seminar

Given a family of complex algebraic varieties and a path in the base, flat connections on the fibres carry an operation of isomonodromic deformation: choosing a path in the base, we can deform a flat connection from one fibre to another along this path while keeping the underlying monodromy representation constant.

We solve the problem of upgrading this operation of isomonodromic deformation along a path to a functor between categories of flat connections with logarithmic singularities along a divisor D. The main tool used is the twisted fundamental groupoid \Pi_1(X,D). As applications, (1) we get that isomonodromy gives a map of moduli stacks of flat connections with logarithmic singularities, (2) we encode higher homotopical information at level 2, i.e. we get an action of the fundamental 2-groupoid of the base of our family on the categories of logarithmic flat connections on the fibres, and (3) our methods produce a geometric incarnation of the isomonodromy functors as Morita equivalences which are more primary than the isomonodromy functors themselves, and from which they can be formally extracted by passing to representation categories.