Glasnik Matematicki, Vol. 61, No. 1 (2026), 117-150. \( \)

THE HITCHIN–KOBAYASHI CORRESPONDENCE FOR QUIVER BUNDLES OVER THE NON-COMPACT AFFINE GAUDUCHON MANIFOLD

Pan Zhang, Meng-Qi Zheng and Chang-Sheng Zhu

School of Mathematical Sciences, Anhui University, Hefei 230601, P.R. China
e-mail:panzhang20100@ahu.edu.cn

School of Mathematical Sciences, Anhui University, Hefei 230601, P.R. China
e-mail:a24201041@stu.ahu.edu.cn

School of Mathematical Sciences, Anhui University, Hefei 230601, P.R. China
e-mail:a24201038@stu.ahu.edu.cn

Abstract.   The objective of this paper is to prove a broader, generalized version of the Hitchin–Kobayashi correspondence for the twisted quiver bundle \(\mathcal{R}\) over the non-compact special affine Gauduchon manifold \((M, D, g, \nu)\). On the one hand, we prove that the analytic \((\sigma,\tau)\)-stability on \(\mathcal{R}\) implies the existence of an affine \((\sigma,\tau)\)-Hermite–Einstein metric. On the other hand, we prove that the analytic \((\sigma,\tau)\)-semi-stability on \(\mathcal{R}\) implies the existence of approximate affine \((\sigma,\tau)\)-Hermite–Einstein structure. The proof of the theorems relies on the heat flow method, alongside the continuity approach by Uhlenbeck and Yau. To overcome the analytical obstacles brought by the structure of the quiver, we use the maximum and minimum values of some eigenvalues to define a new quantity \(\chi\). Based on the method of proof by contradiction, the quantity \(\chi\) can be used in the discussion of constructing weak quiver subbundles that contradict stability or semi-stability.

2020 Mathematics Subject Classification.   53C07, 53A15

Key words and phrases.   Quiver bundle; non-compact affine manifold; analytic \((\sigma,\tau)\)-stability; affine \((\sigma,\tau)\)-Hermite–Einstein

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https://doi.org/10.3336/gm.61.1.05

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