Arithmetic/algebraicity criterion for isomonodromic deformations

Establish, for a logarithmic flat bundle (E,∇) on (\overline{X}_s,D_s) over a smooth base S, the equivalence between (i) existence of an algebraic isomonodromic deformation over S; (ii) existence of an integral formal isomonodromic deformation; and (iii) existence of an ω(p)-integral formal isomonodromic deformation.

Background

This conjecture is the non-linear isomonodromy analogue of the main arithmetic/algebraicity principle, proposing an integrality criterion for algebraic isomonodromic deformations, e.g. for Painlevé VI and Schlesinger systems.

The authors prove the conjecture under Picard–Fuchs initial conditions, but the general equivalence for arbitrary initial conditions remains open and would imply Grothendieck–Katz p-curvature in full generality.