Non-commutative counting and stability

Abstract: The second author and Katzarkov introduced categorical invariants based on counting of full triangulated subcategories in a given triangulated category $\mathcal T$, and they demonstrated different choices of additional properties of the subcategories being counted, in particular - an approach to make non-commutative counting in $\mathcal T$ dependable on a stability condition $\sigma \in {\rm Stab}(\mathcal T)$. In this paper, we focus on this approach. After recalling the definitions of a stable non-commutative curve in $\mathcal T$ and related notions, we prove a few general facts and study an example: $\mathcal T = D^b(Q)$, where $Q$ is the acyclic triangular quiver. In previous papers, it was shown that there are two non-commutative curves of non-commutative genus $1$ and infinitely many non-commutative curves of non-commutative genus $0$ in $D^b(Q)$. Our studies here imply that for an open and dense subset in ${\rm Stab}(D^b(Q))$ the stable non-commutative curves in $D^b(Q)$ are finitely many. This paper also introduces counting of semistable derived points and shows that the corresponding invariants are finite on an open dense subset of ${\rm Stab}\big(D^b(Q)\big)$.