The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper, we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins–Yau and mirror to Thomas' moment map for special Lagrangians, to the Donaldson–Fujiki picture of scalar curvature as a moment map. As a consequence, one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the Kähler–Yang–Mills system of Álvarez–Cónsul, Garcia–Fernandez and García–Prada, and the coupled Kähler–Einstein equations of Hultgren–Witt Nyström. After establishing several general results, we focus on the equations and their large/small radius limits on abelian varieties, with a source term, following ideas of Feng and Székelyhidi.

中文翻译：

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变形 Hermitian Yang-Mills 连接、扩展规范群和标量曲率

变形 Hermitian Yang-Mills (dHYM) 方程是复杂几何中的特殊拉格朗日型条件。它需要使用背景 Kähler 度量为全纯线丛上的陈连接定义的拉格朗日相位的复杂模拟是恒定的。在本文中，我们介绍并研究了具有可变 Kähler 度量的 dHYM 方程。这些是同时涉及拉格朗日相位和半径函数的耦合方程。它们是通过使用扩展规范群将 dHYM 连接的矩图解释耦合来获得的，这是由于柯林斯-丘和托马斯的特殊拉格朗日矩图，以及作为矩图的标量曲率的唐纳森-藤木图。因此，人们期望解应该满足 K 稳定性和布里奇兰型稳定性的混合。在特殊限制或特殊情况下，我们恢复了 Álvarez-Cónsul、Garcia-Fernandez 和 García-Prada 的 Kähler-Yang-Mills 系统，以及 Hultgren-Witt Nyström 的耦合 Kähler-Einstein 方程。在建立了几个一般结果之后，我们关注方程及其对阿贝尔变体的大/小半径限制，并遵循冯和 Székelyhidi 的思想，使用源项。