Bridgeland stability condition with support property

Let $\mathcal{D}$ be an essentially small triangulated category, meaning that it is equivalent to a category whose objects form a set. Write $K(\mathcal{D})$ for its triangulated Grothendieck group, with the distinguished-triangle relation $[B]=[A]+[C]$ for every triangle $A\to B\to C\to A[1]$. Fix a finite-rank free abelian group $\mathrm{\Gamma }$, a group homomorphism $v:K(\mathcal{D})\to \mathrm{\Gamma }$, and a norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ on ${\mathrm{\Gamma }}_{\mathbb{R}}=\mathrm{\Gamma }{\otimes }_{\mathbb{Z}}\mathbb{R}$.

A slicing is a family of full additive subcategories $\mathcal{P}(\varphi )\subseteq \mathcal{D}$, indexed by $\varphi \in \mathbb{R}$, such that $\mathcal{P}(\varphi +1)=\mathcal{P}(\varphi )[1]$, ${\mathrm{Hom}}_{\mathcal{D}}(A_1,A_2)=0$ when $A_i\in \mathcal{P}({\varphi }_i)$ and ${\varphi }_1>{\varphi }_2$, and each nonzero object $E$ has a finite Harder–Narasimhan filtration by distinguished triangles whose nonzero factors belong to $\mathcal{P}({\varphi }_1),\dots ,\mathcal{P}({\varphi }_k)$ with ${\varphi }_1>\cdots >{\varphi }_k$. For $I\subseteq \mathbb{R}$, $\mathcal{P}(I)$ denotes the extension-closed full subcategory generated by the $\mathcal{P}(\varphi )$ with $\varphi \in I$.

A locally finite Bridgeland stability condition with respect to $v$ is a pair $\sigma =(Z,\mathcal{P})$, where $Z:\mathrm{\Gamma }\to \mathbb{C}$ is a group homomorphism and $\mathcal{P}$ is a slicing, satisfying $Z(v(E))=m(E)e^{i\pi \varphi }$ for every nonzero $E\in \mathcal{P}(\varphi )$, with $m(E)>0$, and the source local-finiteness condition: for some $\eta >0$, each $\mathcal{P}((\varphi -\eta ,\varphi +\eta ))$ has finite length in its induced strict exact structure, uniformly in $\varphi$. A nonzero object in some $\mathcal{P}(\varphi )$ is semistable of phase $\varphi$; it is stable when it is simple in that phase subcategory.

The support property means that there is a constant $C>0$ such that $\mathrm{\parallel }v(E)\mathrm{\parallel }\le C\mathrm{\mid }Z(v(E))\mathrm{\mid }$ for every nonzero semistable object $E$. In finite rank this condition is independent of the chosen norm. The stability space ${\mathrm{Stab}}_{\mathrm{\Gamma }}(\mathcal{D})$ consists of these locally finite stability conditions satisfying the support property; no existence of such a condition on an arbitrary category is asserted by this definition.

For a stability condition $\sigma =(Z,\mathcal{P})$, let ${\varphi }_{\sigma }^{+}(E)$ and ${\varphi }_{\sigma }^{-}(E)$ be the largest and smallest phases in its finite Harder–Narasimhan filtration. For two slicings $\mathcal{P},\mathcal{Q}$, set $d(\mathcal{P},\mathcal{Q})={\sup }_{0\mathrm{\ne }E\in \mathcal{D}}\max \{\mathrm{\mid }{\varphi }_{\mathcal{P}}^{+}(E)-{\varphi }_{\mathcal{Q}}^{+}(E)\mathrm{\mid },\mathrm{\mid }{\varphi }_{\mathcal{P}}^{-}(E)-{\varphi }_{\mathcal{Q}}^{-}(E)\mathrm{\mid }\}\mathrm{.}$ For $U\in {\mathrm{Hom}}_{\mathbb{Z}}(\mathrm{\Gamma },\mathbb{C})$, set $\mathrm{\parallel }U{\mathrm{\parallel }}_{\sigma }={\sup }_{E\sigma \text{-semistable},E\mathrm{\ne }0}\mathrm{\mid }U(v(E))\mathrm{\mid }\mathrm{/}\mathrm{\mid }Z(v(E))\mathrm{\mid }$, allowing $+\mathrm{\infty }$. Both suprema are taken in the nonnegative extended real numbers $[0,+\mathrm{\infty }]$, so the supremum of an empty family is $0$. In particular the zero category has the unique zero slicing, $d(\mathcal{P},\mathcal{P})=0$, and $\mathrm{\parallel }U{\mathrm{\parallel }}_{\sigma }=0$ for every charge perturbation $U$; its support condition is vacuous. The source topology on the stability space is generated by $B_{ϵ}(\sigma )=\{(W,\mathcal{Q}):\mathrm{\parallel }W-Z{\mathrm{\parallel }}_{\sigma }<\sin (\pi ϵ),d(\mathcal{P},\mathcal{Q})<ϵ\},0<ϵ<1\mathrm{/}4.$

Triangulated category

Grothendieck group of an essentially small triangulated category