UC Berkeley

In this thesis we develop a framework for understanding certain functors between the derived categories of toric varieties and stacks. We take the viewpoint that, just as toric varieties can be understood by studying commutative monoids, toric stacks can be understood by studying grouplike symmetric monoidal categories (i.e. those whose objects form an abelian group under the symmetric monoidal structure). We apply this perspective to describe the convolution product on the derived category of quasicoherent sheaves on a smooth toric stack [An/G] (viewed as a commutative monoid stack via coordinatewise multiplication). By combining this with Halpern-Leistner’s theory of windows in GIT and with Hanlon–Hicks–Lazarev’s resolutions of the diagonal, we produce descriptions of “quiver tensor products” on certain toric varieties, answering a natural question in homological mirror symmetry. We also construct a novel family of monoidal structures on the derived categories of projective spaces. Finally, we present a sketch of how these ideas can be extended to form a theory of “derived toric geometry” controlling push-pull functors between derived categories of toric varieties and stacks.