It would be nice if they went in to a bit more detail about what they see as the connection between these two subjects.
As they say, a variety is the zero set of some polynomial equations. In particular a Riemann surface can be represented as a polynomial equation P(x,y)=0. This can quantised as you might expect x->x, y-> h d/dx, and you can study the equation P(x,h d/dx) \psi(x) = 0. The solution \psi can be written as a formal power series. For a nice choice of P(x,y), these power series can be a generating function for "something", and can contain a lot of interesting algebraic information.
As they say, a variety is the zero set of some polynomial equations. In particular a Riemann surface can be represented as a polynomial equation P(x,y)=0. This can quantised as you might expect x->x, y-> h d/dx, and you can study the equation P(x,h d/dx) \psi(x) = 0. The solution \psi can be written as a formal power series. For a nice choice of P(x,y), these power series can be a generating function for "something", and can contain a lot of interesting algebraic information.