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Contractibility and total semi-stability conditions of Euclidean quivers

Yu Qiu, Xiaoting Zhang

Abstract

We study the bounded derived category $\mathcal{D}$ of an Euclidean quiver, or equivalently, that of coherent sheaves on a tame weighted projective line. We give a description of the moduli space $\mathrm{ToSS}$ of the total semi-stability conditions on $\mathcal{D}$, which implies that $\mathrm{ToSS}$ can linearly contract to any chosen non-concentrated stability condition in it. For type $\widetilde{A_{p,q}}$, this gives an alternative proof of the contractibility of the whole space of stability conditions.

Contractibility and total semi-stability conditions of Euclidean quivers

Abstract

We study the bounded derived category of an Euclidean quiver, or equivalently, that of coherent sheaves on a tame weighted projective line. We give a description of the moduli space of the total semi-stability conditions on , which implies that can linearly contract to any chosen non-concentrated stability condition in it. For type , this gives an alternative proof of the contractibility of the whole space of stability conditions.

Paper Structure

This paper contains 21 sections, 14 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Stability conditions
  3. Total semi-stability
  4. Preliminaries
  5. Stability conditions on triangulated categories
  6. Total semi-stability
  7. Weighted projective lines of tame type and canonical algebras
  8. Auslander-Reiten (AR) quiver
  9. Tubes in $\operatorname{AR}\operatorname{coh}_0(\mathbb{P}^1_\mathbf{w})$
  10. Partitions of the imaginary root
  11. Total semi-stability data: the prototype
  12. The standard heart
  13. Case I
  14. Case II
  15. Total semi-stability datum
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\operatorname{\mathbb{P} ToSS}(\mathbb{P}^1_\mathbf{w})$ be the projective space of total semi-stable stability conditions on $\operatorname{\mathcal{D}}^b(\mathbb{P}^1_\mathbf{w})$.

Figures (4)

  • Figure 1: Part of the AR-quiver of $\operatorname{\mathcal{D}}^b(\mathbb{P}^1_\mathbf{w})\cong\operatorname{\mathcal{D}}(\operatorname{C}_\mathbf{w})$: type $\widetilde{A_{3,2}}$
  • Figure 2: Part of the AR-quiver of $\operatorname{\mathcal{D}}(\operatorname{C}_\mathbf{w})$: type $\widetilde{D_6}$
  • Figure 3: The canonical algebra quiver in the AR-quiver of $\operatorname{\mathcal{D}}(\operatorname{C}_\mathbf{w})$: type $\widetilde{E_6}$
  • Figure 4: The canonical algebra quivers in the AR-quivers of $\operatorname{\mathcal{D}}(\operatorname{C}_\mathbf{w})$: types $\widetilde{E_7}$ and $\widetilde{E_8}$

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 19 more