发表机构
University of Haifa（海法大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明任意完备非阿基米德域上有限箭图的每个多稳定表示均存在HKKP意义下的分裂调和范数，并给出扩展HKKP势的最小值达成性及分裂范数的解析族分离结果。
AI 中文摘要
设$K$为任意完备非阿基米德域，不一定是球完备的。我们证明，$K$上有限箭图的每个多稳定表示都容许一个Haiden--Katzarkov--Kontsevich--Pandit（HKKP）意义下的分裂调和范数，回答了他们的预印本《走向范畴Kähler几何》中的一个问题。Moret-Bailly的闭像定理给出了与具有去稳定子对象的表示的均匀分离。对范数权重的有限估计进而得出稳定表示的线性强制界，从而在所有范数的完备CAT(0)空间上得到扩展HKKP势的最小值。直接的一阶变分计算将分裂极小元与HKKP调和范数等同起来。严格支撑铰链将分裂达成归结为具有任意实数半径标签的箭图的半稳定约化。提升分裂的小变化使每个反向扩展非分裂；有限斜率--秩下降证明终止性。因此，完备范数空间中的每个最小值都由具有相同箭尾奇异轮廓的分裂范数达成，对值群或箭图没有限制。非分裂极小元说明了一次和两次半径修改中的分裂达成，而一个双参数Berkovich解析族分离了所选分裂范数的多稳定性、半稳定性和调和性。
英文摘要
Let $K$ be any complete non-Archimedean field, not necessarily spherically complete. We prove that every polystable representation of a finite quiver over $K$ admits a split harmonic norm in the sense of Haiden--Katzarkov--Kontsevich--Pandit (HKKP), answering a question from their preprint \emph{Towards Categorical Kähler Geometry}. Moret-Bailly's closed-image theorem gives a uniform separation from representations with destabilizing subobjects. A finite estimate on norm weights then yields a linear coercivity bound for stable representations and hence a minimum of the extended HKKP potential on the complete CAT(0) space of all norms. A direct first-variation computation identifies split minimizers with HKKP harmonic norms. Strict supporting hinges reduce split attainment to semistable reduction for quivers with arbitrary real radius labels. Small changes of the lifted splitting make each reversed extension nonsplit; a finite slope--rank descent proves termination. Consequently, every minimum in the completed norm space is attained by a split norm with the same arrow singular profiles, without restrictions on the value group or the quiver. Nonsplit minimizers illustrate split attainment in one and two radius modifications, and a two-parameter Berkovich analytic family separates polystability, semistability, and harmonicity of a chosen split norm.
Comments39 pages, comments welcome
