  In this manuscript we study braid varieties, a class of affine algebraic
varieties associated to positive braids. Several geometric constructions are
presented, including certain torus actions on braid varieties and holomorphic
symplectic structures on their respective quotients. We also develop a
diagrammatic calculus for correspondences between braid varieties and use these
correspondences to obtain interesting stratifications of braid varieties and
their quotients. It is shown that the maximal charts of these stratifications
are exponential Darboux charts for the holomorphic symplectic structures, and
we relate these strata to exact Lagrangian fillings of Legendrian links.
