Optimal lower bounds for Donaldson's J-functional

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 Kähler (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 Kähler cone, even when the underlying Kähler classes admit cscK metrics. As a final application we show that if the Lejmi-Székelyhidi 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 Kähler cone, and there exists an optimal degeneration for uniform J-stability.

arXiv commentFinal version, to appear in Adv. Math

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• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Existence of cscK metrics on smooth minimal models
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• Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow
• Openness of uniform K-stability in the Kähler cone
• A decomposition formula for J-stability and its applications
• Constant scalar curvature Kaehler metrics on ramified Galois coverings
• Minimal slopes and bubbling for complex Hessian equations
• Twisted and coupled constant scalar curvature Kähler metrics on minimal ruled surfaces