So I was surprised to find that in recent years computers have been calculating de Rham cohomology, and doing so using methods from algebraic geometry. In fact, functions to compute the de Rham cohomology of the complement of various algebraic varieties have been available in Macaulay2 for a while. There's a chapter of the Macaulay2 book available on the subject. The methods used are based on D-modules and discussed in the book Grobner Deformations of Hypergeometric Differential Equations.
I have no idea where the connection with hypergeometric series is coming from but it's interesting to see the connection with algebraic geometry. D-modules are rings with elements xi and di sith the proprty that xixj=xjxi and didj=djdi but that dixj-xjdi=δij. We can form ideals of these things and study them using Grobner basis methods. Given a smooth function we can form the ideal of elements of a D-module that annihilate the function by considering the di to be the differential operators d/dxi. This gives a connection between the methods of algebraic geometry and differential geometry. But that's as far as my knowledge extends for now.