Doctor of Philosophy (PhD)
In [BGS96], Beilinson, Ginzburg, and Soergel introduced the notion of mixed categories. This idea often underlies many interesting "Koszul dualities." In this paper, we produce a mixed derived category of constructible complexes (in the sense of [BGS96]) for any toric variety associated to a fan. Furthermore, we show that it comes equipped with a t-structure whose heart is a mixed version of the category of perverse sheaves. In chapters 2 and 3, we provide the necessary background. Chapter 2 concerns the categorical preliminaries, while chapter 3 gives the background geometry. This concerns both some basics of toric varieties as well as basics of constructible sheaves in this setting. In chapter 4, we introduce the primary category of interest, Dmix(X0) for a toric variety X0 defined over some finite field. We prove that this is a mixed version of Dbc(X), the bounded derived category of constructible complexes over X = X0 xSpec(Fq) Spec(F q), the variety obtained by extension of scalars. In chapter 5, we introduce the standard suite of functors associated to a locally closed inclusion of toric varieties, h : Y0 → X0, between the mixed categories Dmix(X0) and Dmix(Y0). We provide some functors associated to other special types of toric maps as well. Finally, we prove that some of these functors commute with the realization functor r : Dmix(X0) → DbT,m(X0). We call this being genuine.
Taylor, Sean Michael, "Mixed Categories of Sheaves on Toric Varieties" (2018). LSU Doctoral Dissertations. 4590.