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The geometric Merkurjev-Panin Conjecture for the Cox category

Daniel Erman, Andrew Hanlon, Gaku Liu, Hailun Zheng

TL;DR

This paper addresses a geometric Merkurjev-Panin question for smooth projective toric varieties by working in the Cox category. The authors construct a $G$-action on $D_{ ext{Cox}}(X)$ compatible with toric automorphisms and prove that the Bondal-Thomsen collection $\Theta$ is $G$-invariant, yielding a $G$-invariant tilting bundle and, in the projective case, a $G$-invariant full strong exceptional collection. Consequently, $K_0(X)$ embeds as a direct summand of the permutation $G$-module $K_{ ext{Cox}}(X)$, reinforcing the philosophy that $D_{ ext{Cox}}(X)$ captures richer symmetry than the derived category of the variety itself. The work extends to semiprojective and simplicial contexts and includes a detailed exploration of the permutohedral variety as a representative example of the resulting combinatorial richness.

Abstract

We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exceptional collection of Bondal-Thomsen line bundles is invariant under the group of lattice automorphisms that permute the rays of the toric variety's fan. Our result is meant to further illustrate that the Cox category is a natural repository for homological algebra on toric varieties.

The geometric Merkurjev-Panin Conjecture for the Cox category

TL;DR

This paper addresses a geometric Merkurjev-Panin question for smooth projective toric varieties by working in the Cox category. The authors construct a -action on compatible with toric automorphisms and prove that the Bondal-Thomsen collection is -invariant, yielding a -invariant tilting bundle and, in the projective case, a -invariant full strong exceptional collection. Consequently, embeds as a direct summand of the permutation -module , reinforcing the philosophy that captures richer symmetry than the derived category of the variety itself. The work extends to semiprojective and simplicial contexts and includes a detailed exploration of the permutohedral variety as a representative example of the resulting combinatorial richness.

Abstract

We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exceptional collection of Bondal-Thomsen line bundles is invariant under the group of lattice automorphisms that permute the rays of the toric variety's fan. Our result is meant to further illustrate that the Cox category is a natural repository for homological algebra on toric varieties.
Paper Structure (7 sections, 13 theorems, 21 equations, 3 figures)

This paper contains 7 sections, 13 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background and setup
  4. Extending the equivariant structure to the Cox category
  5. Proof of Theorem \ref{['thm:main']}
  6. Further invariant structures
  7. Example: permutohedral variety

Key Result

Theorem A

The Bondal-Thomsen collection is a $G$-invariant full strong exceptional collection for $D_{\operatorname{Cox}}(X)$. In particular, if we write $K_{\operatorname{Cox}}(X)$ for the Grothendieck group of $D_{\operatorname{Cox}}(X)$, then $K_{\operatorname{Cox}}(X)$ is a permutation $G$-module with the

Figures (3)

  • Figure 5.1: Examples of $S_\theta$ and $P_\theta$ drawn on the periodic hyperplane arrangement on $M_\mathbb{R}$ that lifts the Bondal stratification for $\mathbb P^2$ with $\theta_1$ and $\theta_2$ corresponding to ${\mathcal{O}}(-1)$ and ${\mathcal{O}}(-2)$, respectively.
  • Figure 6.1: A fundamental domain of the Bondal stratification for the permutohedral variety $X_3$ with the strata labeled by the corresponding Bondal-Thomsen elements.
  • Figure 6.2: On the left, we cut the cube into two simplices and an octahedron as a first step in understanding the Bondal stratification for $X_4$. On the right, we finish drawing the stratification by cutting the octahedron into eight simplices.

Theorems & Definitions (26)

  • Theorem A
  • Corollary B
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop:ExtendingEquivariance']}(1)
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Proposition \ref{['prop:ExtendingEquivariance']}(2)
  • proof : Proof of Proposition \ref{['prop:ExtendingEquivariance']}(3)
  • ...and 16 more