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On the derived category of a toric stack bundle

Qian Chao, Jiun-Cheng Chen, Hsian-Hua Tseng

TL;DR

This paper analyzes the $\bold{T}$-equivariant derived category of split toric stack bundles over a smooth projective base $B$ by constructing explicit building blocks and functors. It provides (i) a strong full exceptional collection on the toric fiber, (ii) spanning classes for the bundle, (iii) a semi-orthogonal decomposition of $D^b_{\bold{T}}(\mathfrak{E})$ built from pullbacks of $D^b(B)$ and toric data, and (iv) a Fourier–Mukai equivalence for crepant wall-crossings extended to the bundle setting. The results generalize Bondal–Orlov type decompositions to toric stack bundles and establish derived equivalences in geometric transitions, enhancing understanding of equivariant derived categories in toric geometry with potential applications to birational geometry and mirror symmetry.

Abstract

We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.

On the derived category of a toric stack bundle

TL;DR

This paper analyzes the -equivariant derived category of split toric stack bundles over a smooth projective base by constructing explicit building blocks and functors. It provides (i) a strong full exceptional collection on the toric fiber, (ii) spanning classes for the bundle, (iii) a semi-orthogonal decomposition of built from pullbacks of and toric data, and (iv) a Fourier–Mukai equivalence for crepant wall-crossings extended to the bundle setting. The results generalize Bondal–Orlov type decompositions to toric stack bundles and establish derived equivalences in geometric transitions, enhancing understanding of equivariant derived categories in toric geometry with potential applications to birational geometry and mirror symmetry.

Abstract

We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.
Paper Structure (8 sections, 4 theorems, 32 equations)

This paper contains 8 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Constructions
  3. Organization
  4. Acknowledgement
  5. Exceptional objects
  6. Spanning classes
  7. Semi-orthogonal decomposition
  8. Fourier-Mukai equivalence

Key Result

Lemma 3.1

The collection of $\bold{T}$-equivariant line bundles is a spanning class of $D^b_{\bold{T}}(\mathfrak{X})$.

Theorems & Definitions (8)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Proposition 5.1
  • proof