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Harmonic norms on non-Archimedean quiver representations

Does every polystable representation of a finite quiver over a complete non-Archimedean field admit a split harmonic norm in the sense of Haiden–Katzarkov–Kontsevich–Pandit (HKKP)? The target allows fields that are not spherically complete and imposes no value-group restriction. The trivially valued case is included, although existence there was already known.

Why it matters and what counts as progress

Why it matters: Existence in this generality would remove field-completeness restrictions from a norm-construction problem in categorical geometry.

What counts as progress: Prove split harmonic-norm existence for a broader field class or for every field in the stated setting, or exhibit a polystable counterexample. Existence of a nonsplit minimizer alone does not establish attainment by the required split norm.

How the problem developed

  1. 1 Sept 2026 · Open question [1]

    Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich and Pranav Pandit Asked whether spherical completeness can be removed from their theorem that polystable quiver representations admit harmonic norms.

  2. 1 Oct 2026 · Theorem [1][2]

    Oren Ben-Bassat Reported that every polystable representation of a finite quiver over any complete non-Archimedean field admits a split harmonic norm, answering the question without spherical completeness.

AI contribution

Sources

  1. Haiden, Katzarkov, Kontsevich and Pandit, Towards Categorical Kähler Geometry
  2. Oren Ben-Bassat, Harmonic Norms on Quiver Representations over Non-Archimedean Fields
  3. Original paper and AI contribution disclosure
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