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On the Yau-Tian-Donaldson Conjecture for Generalized Kähler-Ricci Soliton Equations
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2022-05-14 , DOI: 10.1002/cpa.22053
Jiyuan Han 1 , Chi Li 2
Affiliation  

Let (X,D) be a polarized log variety with an effective holomorphic torus action, and Θ be a closed positive torus invariant (1,1) -current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci g-solitons. We prove a version of the Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler-Ricci/Mabuchi solitons or Kähler-Einstein metrics. © 2022 Wiley Periodicals, Inc.

中文翻译:

关于广义 Kähler-Ricci 孤子方程的 Yau-Tian-Donaldson 猜想

令 (X,D) 为具有有效全纯环面作用的极化对数变量,θ 为闭合正环面不变量 (1,1) -电流。对于在环面作用的矩多胞形上定义的任何平滑正函数g ,我们研究了对应于广义和扭曲的 Kähler-Ricci g孤子的 Monge-Ampère 方程。我们证明了这些一般方程的 Yau-Tian-Donaldson (YTD) 猜想的一个版本,表明解的存在性总是等价于等变均匀的 θ - 扭曲g -Ding稳定性。当θ是与环面不变线性系统相关的电流,我们进一步证明等变特殊测试配置足以测试稳定性。我们的结果允许任意 klt 奇点,并概括了(扭曲)Kähler-Ricci/Mabuchi 孤子或 Kähler-Einstein 度量的(统一)YTD 猜想的大部分先前结果。© 2022 Wiley 期刊公司。
更新日期:2022-05-14
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