Representing boolean functions by polynomials or rational functions either perfectly or approximately is an important topic while polynomials and rational functions is the body of algebraic geometry. Why is algebraic geometry not that applicable to this area of complexity theory? Or atleast what are some of the important papers in representing boolean functions that uses algebraic geometry?
In addition to the Geometric Complexity Theory Program already mentioned by Sasho Nikolov (see e.g. here, and - shameless self plug, but has tons of references on uses of AG in complexity - here), there's also:
Work on matrix multiplication (Strassen's work especially come to mind, as well as the more recent GCT-style work of Burgisser and Ikenmeyer, but also some other stuff in between)
Mulmuley's "P vs NC" result (this is a Boolean result - if you need convincing I'm happy to), which can be seen as building off of Ben-Or's sorting lower bound and others
Work on derandomizing polynomial identity testing (several papers of M. Agrawal come to mind). (Which, in retrospect, is deeply related to GCT.)
Coding theory (there's a whole "subfield" on AG codes)