Good formal structures for flat meromorphic connections, III: Towards functorial modifications

Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, we proved existence of good formal structures and a good DeligneMalgrange lattice after suitably blowing up. For the corresponding situation over a complex analytic space, one immediately obtains the existence of suitable blowups locally, but it is not clear that these blowups can be glued together. We outline an approach to constructing a global blowup by making the result for excellent schemes functorial for regular morphisms. However, the approach remains conditional on resolution of a problem of birational geometry (canonical determination of nef Cartier b-divisors).