King's Conjecture and the Cox category

TL;DR

This work constructs the Cox category $D_{ ext{Cox}}(X)$ by gluing derived categories of all toric models arising from a fixed Cox ring, thereby realizing a broad version of King's Conjecture for semiprojective toric varieties. Central to the approach is the Bondal–Thomsen collection $\Theta$, which is shown to be tilting for $D_{ ext{Cox}}(X)$ (and a full strong exceptional collection when $X$ is projective), with a proof that relies on a delicate $\Theta$-Transform Lemma governing Fourier–Mukai transforms across GKZ chambers. The paper further develops a uniform diagonal (HHL) resolution in $D_{ ext{Cox}}$, establishes window categories and noncommutative resolutions arising from the Cox data, and provides monadic and sharpened generation frameworks tied to the GKZ geometry. Collectively, these results unify Beilinson–type tilting, Bondal–Thomsen generation, and GKZ-driven birational geometry in a single invariant category, with broad algebraic and symplectic/mirror-symmetric implications across all toric models tied to a given Cox ring.

Abstract

We state and prove a realization of King's Conjecture for a category glued from the derived categories of all of the toric varieties arising from a given Cox ring. Our perspective extends ideas of Beilinson and Bondal to all semiprojective toric varieties.

Figures (6)

• Figure 1.1: The secondary fan for a Hirzebruch surface $\mathcal{H}_3$ has two maximal chambers: one corresponding to $\mathcal{H}_3$ and the other to $\mathbb P(1,1,3)$. The Bondal--Thomsen collection $\Theta$ consists of the degrees in a half-open zonotope $Z$ determined by the degrees of the variables. In this example, $\Theta$ consists of the $6$ degrees $-d_i$, with $d_i$ as marked in $-Z$.
• Figure 1.2: The pictures at left and right indicate the boundary conditions for Lagrangians in the cotangent bundle of a torus, drawn as partial conormals to a stratification of the torus, in the correspondence between derived categories of toric varieties and wrapped Fukaya categories. The union of this "stop data" yields a Fukaya category with no obvious algebraic analogue. This motivated our definition of the Cox category.
• Figure 4.1: The GKZ fan from Example \ref{['ex:Bl2points']} has five maximal chambers. When $n=3$, we label each chamber and face with the corresponding variety.
• Figure 4.2: This figure illustrates some of the polytopes that appear in the proof of \ref{['lem:thetatransform2']} in the case that ${\mathcal{X}}_i=\mathcal{H}_3$ is the Hirzebruch surface $\mathcal{H}_3$ from \ref{['fig:HirzSecondaryFan']}. See \ref{['ex:hirzPolytopes']} for a detailed description.
• Figure 4.3: This figure demonstrates a sample computation of $\operatorname{Hom}({\mathcal{O}}_{\operatorname{Cox}}(-d),{\mathcal{O}}_{\operatorname{Cox}}(-d'))$. See \ref{['ex:CoxHoms']} for a detailed description.

Theorems & Definitions (132)

• Theorem A
• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5: $\Theta$-Transform Lemma
• Definition 2.2: Cox construction
• Definition 2.3
• Proposition 2.6
• Proposition 2.7