Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global dimension of the derived category of an orbifold projective line

Takumi Otani

TL;DR

This work determines the global dimension of the derived category ${\mathcal D}^b({\mathbb P}^1_{A,\Lambda})$ of an orbifold projective line by analyzing stability conditions and slope stability across the three Euler characteristic types $\chi_A>0$, $\chi_A=0$, and $\chi_A<0$. It shows that ${\rm gldim}$ attains the value 1 in the domestic and tubular cases (via slope-stability and fractional Calabi–Yau structure) and remains strictly greater than 1 for any stability condition in the wild case, while the overall derived category always has ${\rm gldim}=1$ due to approximation by near-1 gldim stability conditions. The results connect to Serre and upper Serre dimensions, Gepner-type phenomena, and known quiver descriptions (via derived equivalences to extended Dynkin quivers) and align with the conformal dimensions from genus-zero Gromov–Witten theory. Overall, the paper integrates stability conditions, Serre theory, and orbifold geometry to classify the global dimension landscape for these triangulated categories.

Abstract

The global dimension of a triangulated category is defined to be the infimum value of the global dimensions of stability conditions on the triangulated category. In this paper, we study the global dimension of the derived category of an orbifold projective line.

Global dimension of the derived category of an orbifold projective line

TL;DR

This work determines the global dimension of the derived category of an orbifold projective line by analyzing stability conditions and slope stability across the three Euler characteristic types , , and . It shows that attains the value 1 in the domestic and tubular cases (via slope-stability and fractional Calabi–Yau structure) and remains strictly greater than 1 for any stability condition in the wild case, while the overall derived category always has due to approximation by near-1 gldim stability conditions. The results connect to Serre and upper Serre dimensions, Gepner-type phenomena, and known quiver descriptions (via derived equivalences to extended Dynkin quivers) and align with the conformal dimensions from genus-zero Gromov–Witten theory. Overall, the paper integrates stability conditions, Serre theory, and orbifold geometry to classify the global dimension landscape for these triangulated categories.

Abstract

The global dimension of a triangulated category is defined to be the infimum value of the global dimensions of stability conditions on the triangulated category. In this paper, we study the global dimension of the derived category of an orbifold projective line.
Paper Structure (10 sections, 14 theorems, 35 equations)

This paper contains 10 sections, 14 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability condition
  4. Global dimension of a triangulated category
  5. Orbifold projective line
  6. Slope stability
  7. Global dimension of the derived category
  8. Proof of Theorem \ref{['thm : main 2']} for the case $\chi_A > 0$
  9. Proof of Theorem \ref{['thm : main 2']} for the case $\chi_A = 0$
  10. Proof of Theorem \ref{['thm : main 2']} for the case $\chi_A < 0$

Key Result

Theorem 1.1

Let $\sigma$ be a stability condition on ${\mathcal{D}}^b({\mathbb P}_{A, \Lambda}^1)$. For $\tau \in {\mathbb H}$, define a homomorphism $Z_\tau \colon K_0({\rm coh}({\mathbb P}_{A, \Lambda}^1)) \longrightarrow {\mathbb C}$ by We have $\sigma = (Z_\tau, {\rm coh}({\mathbb P}_{A, \Lambda}^1))$ for some $\tau \in {\mathbb H}$ if and only if $\sigma$ satisfies the following conditions:

Theorems & Definitions (29)

  • Theorem 1.1: Theorem \ref{['thm : main 1']}
  • Theorem 1.2: Theorem \ref{['thm : main 2']}
  • Definition 2.1: B
  • Definition 2.2: T
  • Definition 2.3: IQ
  • Definition 2.4: Q
  • Proposition 2.5: KOT
  • Proposition 2.6: Q
  • Definition 3.1: GL
  • Definition 3.2
  • ...and 19 more