Let Q be a Dynkin quiver, let D^b(Q) denote its bounded derived category, and let sigma be a stability condition in Stab(D^b(Q)). A full sigma-exceptional collection is an exceptional collection that generates D^b(Q) and is compatible with sigma. Dynkin stability-exceptional collection conjecture. For each Dynkin quiver Q and each sigma ∈ Stab(D^b(Q)) there exists a full sigma-exceptional collection. This extends the established regularity-preserving results for representation categories of Dynkin quivers; the existence of such collections for every stability condition is proposed as a direction for future research.

Full stability-exceptional collections for Dynkin quivers

The conjecture, attributed in the literature to Dimitrov–Katzarkov, asserts that every stability condition on the derived category of a Dynkin quiver admits a full compatible exceptional collection.

Known results

The affine $A^{(1)}_2$ case was known before the general Dynkin statement; the 2014 literature records the Dynkin case as unresolved.

2022 claimed proof and 2025 extension

The 2022 paper Full exceptional collections and stability conditions for Dynkin quivers claims that every $\sigma in \mathrm{Stab}(D^b(Q))$ has a full monochromatic $\sigma$ -exceptional collection, whose objects are the simple objects of the relevant heart; it identifies this as an affirmative answer to Dimitrov–Katzarkov Conjecture 7.1. A 2025 preprint claims the stronger result for every finite acyclic quiver, hence also for Dynkin quivers. The argument uses generation of hearts by iterated simple tilts.

Current status (as of August 2026): The conjecture is claimed solved by the 2022 result and strengthened by the 2025 preprint, but these claims remain unverified in the retrieved record.

Full stability-exceptional collections for Dynkin quivers

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Progress summary

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