[Paper Review] T-stabilities for a weighted projective line

Shiquan Ruan, Xintian Wang|arXiv (Cornell University)|Oct 3, 2017

Advanced Differential Geometry Research12 references3 citations

TL;DR

This paper establishes a sufficient and necessary condition for finest t-stabilities in piecewise hereditary triangulated categories, and applies it to the bounded derived category of coherent sheaves on a weighted projective line of weight type (2). It proves the existence of a t-exceptional triple, which implies the existence of a σ-exceptional triple for any stability condition σ on the acyclic triangular quiver, thereby confirming the connectedness of the space of stability conditions.

ABSTRACT

The present paper focuses on the study of t-stabilities on a triangulated category in the sense of Gorodentsev, Kuleshov and Rudakov. We give an equivalent description for the finest t-stability on a piecewise hereditary triangulated category and, describe the semistable subcategories and final HN triangles for (exceptional) coherent sheaves in $D^b( m{coh}\mathbb{X})$, which is the bounded derived category of coherent sheaves on the weighted projective line $\mathbb{X}$ of weight type (2). Furthermore, we show the existence of a t-exceptional triple for $D^b( m{coh}\mathbb{X})$. As an application, we obtain a result of Dimitrov--Katzarkov which states that each stability condition $σ$ in the sense of Bridgeland admits a $σ$-exceptional triple for the acyclic triangular quiver $Q$. Note that this implies the connectedness of the space of stability conditions associated to $Q$.

Motivation & Objective

• To characterize finest t-stabilities in piecewise hereditary triangulated categories.
• To describe semistable subcategories and final HN triangles for coherent sheaves in $D^b({\rm coh\,}{\mathbb{X}})$ for a weighted projective line $\mathbb{X}$ of weight type (2).
• To introduce and establish the existence of t-exceptional sequences, particularly a t-exceptional triple, in $D^b({\rm coh\,}{\mathbb{X}})$.
• To apply the t-exceptional triple result to recover and strengthen a result by Dimitrov–Katzarkov on the connectedness of the space of stability conditions for the acyclic triangular quiver.

Proposed method

• The authors define a finest t-stability on a triangulated category and derive a necessary and sufficient condition for a t-stability to be finest in the piecewise hereditary case.
• They analyze the structure of semistable subcategories and final HN (Harder-Narasimhan) triangles for coherent sheaves in $D^b({\rm coh\,}{\mathbb{X}})$, where $\mathbb{X}$ is a weighted projective line of weight type (2).
• The paper introduces the concept of a t-exceptional sequence and proves the existence of a t-exceptional triple in $D^b({\rm coh\,}{\mathbb{X}})$.
• Using the equivalence between $D^b({\rm coh\,}{\mathbb{X}})$ and $D^b({\rm mod}\,kQ)$ for the acyclic triangular quiver $Q$, the authors lift the t-exceptional triple to a σ-exceptional triple for any stability condition σ on $D^b({\rm mod}\,kQ)$.
• They apply results from Macri and Dimitrov–Katzarkov on exceptional sequences and mutation to show that the existence of a σ-exceptional triple implies the connectedness of the space of stability conditions.

Experimental results

Research questions

• RQ1 What is a necessary and sufficient condition for a t-stability to be finest in a piecewise hereditary triangulated category?
• RQ2 How can the semistable subcategories and final HN triangles be explicitly described for coherent sheaves in $D^b({\rm coh\,}{\mathbb{X}})$ when $\mathbb{X}$ has weight type (2)?
• RQ3 Does a t-exceptional triple exist in $D^b({\rm coh\,}{\mathbb{X}})$ for a weighted projective line of weight type (2)?
• RQ4 Can the existence of a t-exceptional triple be used to deduce the existence of a σ-exceptional triple for any stability condition σ on the derived category of the acyclic triangular quiver?

Key findings

• A necessary and sufficient condition is established for a t-stability to be finest in a piecewise hereditary triangulated category.
• For $D^b({\rm coh\,}{\mathbb{X}})$ with $\mathbb{X}$ of weight type (2), the semistable subcategories and final HN triangles for coherent sheaves are explicitly described.
• A t-exceptional triple exists in $D^b({\rm coh\,}{\mathbb{X}})$, which is a key structural result.
• The existence of a t-exceptional triple implies the existence of a σ-exceptional triple for any stability condition σ on $D^b({\rm mod}\,kQ)$ for the acyclic triangular quiver $Q$, as shown via the equivalence of categories.
• This result confirms the connectedness of the space of stability conditions on $D^b({\rm mod}\,kQ)$, recovering a key result of Dimitrov–Katzarkov.
• The paper provides a new proof of the connectedness of the stability manifold for the acyclic triangular quiver using t-stability and t-exceptional sequences.