The Differential Galois Group of One-parametric Systems of Linear Differential Equations
Phùng Hô Hai
(joint work with Nguyen Dai Duong and Joao Pedro dos Santos)
Institute of Mathematics, Vietnam Academy of Science and Technology
On a projective complex manifold $X$, the Riemann-Hilbert correspondence associates to a system of linear differential equations on $X$ a representation of the topological fundamental group of $X$ (at a fixed base point), called the monodromy representation. The image of this representation is called the monodromy group of the system. By Serre’s GAGA principle, this correspondence can be constructed in a pure algebraic way for smooth projective schemes over the complex numbers. The algebraic analog of the monodromy group is the differential Galois group, defined by applying Tannakian duality to the tensor abelian category of all systems of linear differential equations on $X$ (i.e. the category of flat connections on $X$). The construction is further developed for proper smooth schemes in positive characteristic, in which flat connections are replaced by stratified bundles. The study of the differential Galois groups has attracted attention of many authors since more than 40 years. Most notable is the Grothendieck-Katz p-curvature conjecture.
The aim of our work is to study the differential Galois group of an infinitesimal one-parametric family of proper schemes. Thus let $X$ be a proper scheme upon $S= Spec R$, where $R$ is a complete discrete valuation ring. Our approach is to utilize Tannakian duality to construct the differential Galois group as an affine flat group scheme over $R$. We study this group scheme and deduce from it some information about the family of systems of differential equations.