中岛箭图簇、箭图链和星形箭图的比亚林斯基 - 比鲁拉分解
Białynicki-Birula Decompositions of Nakajima Quiver Varieties, Quiver Chains and Star-Shaped Quivers
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中文总结 AI 辅助
研究中岛箭图簇相关的比亚林斯基 - 比鲁拉分解，通过箭图链描述不动点，探讨分解几何，如吸引纤维维度等，还将框架用于星形箭图，对特定维数向量分类不动点轨迹分量并计算动机类。
中文摘要 AI 辅助
本文研究了与中岛箭图簇上自然\(\mathbb{C}^*\)作用相关的比亚林斯基 - 比鲁拉分解，该作用通过沿加倍箭图的反向箭头缩放映射给出。此作用的不动点用辅助箭图（称为箭图链）的带关系表示来描述。还研究了所得比亚林斯基 - 比鲁拉分解几何的几个方面，包括吸引纤维的维度以及中岛箭图簇根据箭图链模空间的相应动机分解。最后，将一般框架专门应用于星形箭图，对特定维数向量对不动点轨迹分量进行分类，并在簇的格罗滕迪克环的合适局部化中计算完整中岛箭图簇的动机类。
英文摘要
This paper studies the Białynicki--Birula decomposition associated with the natural $\mathbb C^*$-action on Nakajima quiver varieties given by scaling the maps along the reversed arrows of the doubled quiver. Fixed points of this action are described in terms of representations with relations of auxiliary quivers, called quiver chains. Several aspects of the geometry of the resulting Białynicki--Birula decomposition are also investigated, including the dimensions of their attracting fibers and the corresponding motivic decomposition of the Nakajima quiver variety in terms of quiver-chain moduli spaces. Finally, the general framework is specialized to star-shaped quivers, where, for particular dimension vectors, the fixed-locus components are classified and the motivic class of the full Nakajima quiver variety is computed in a suitable localization of the Grothendieck ring of varieties.
