We will use symplectic geometry to prove some new results on the algebraic geometry of toric varieties. Namely, we will revisit ideas of Bondal for viewing the toric Frobenius morphism through the lens of homological mirror symmetry. This perspective and recent structural results for Fukaya categories allow us to produce a natural resolution of the structure sheaf of any toric subvariety in a smooth toric variety. In fact, this resolution is built from a finite set of line bundles and has length equal to the codimension of the subvariety, which implies, in particular, that the Rouquier dimension of the derived category of any toric variety coincides with the dimension of the variety. This is joint work in progress with Jeff Hicks and Oleg Lazarev.
