Abstract In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution, and the explicit formula has a number of applications. First, this leads to new existence criteria for constant scalar curvature Kahler (cscK) metrics in terms of Tian's alpha invariant. Moreover, we use the above formula to discuss Calabi dream manifolds and an analogous notion for the J-equation, and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the Kahler cone, even when the underlying Kahler classes admit cscK metrics. As a final application we show that if the Lejmi-Szekelyhidi conjecture holds, then the optimal bound coincides with its algebraic counterpart, the set of J-semistable classes equals the closure of the set of uniformly J-stable classes in the Kahler cone, and there exists an optimal degeneration for uniform J-stability.

中文翻译：

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唐纳森 J 泛函的最佳下限

摘要 在本文中，我们提供了唐纳森 J 泛函的最优下界的明确公式，在明确找到矫顽力定义中的最优常数的意义上，该常数始终存在并且一般取负值。如果 J 方程允许解，那么这个常数正好是正的，并且显式公式有许多应用。首先，这导致了根据田的 alpha 不变量的恒定标量曲率 Kahler (cscK) 度量的新存在标准。此外，我们使用上述公式来讨论 Calabi 梦流形和 J 方程的类似概念，并表明对于曲面，最优边界是一个显式可计算的有理函数，当底层类接近边界时，该函数通常趋于负无穷大卡勒锥体，即使底层 Kahler 类承认 cscK 指标。作为最后的应用，我们证明如果 Lejmi-Szekelyhidi 猜想成立，则最优边界与其代数对应物一致，J-半稳定类的集合等于 Kahler 锥中一致 J-稳定类集合的闭包，并且存在均匀 J 稳定性的最佳退化。