Applications of algebraic geometry in Boolean complexity
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)