# Optimal lower bounds for Donaldson's J-functional

## TL;DR

This paper derives an explicit formula for the optimal lower bound of Donaldson's J-functional, linking it to the existence of solutions to the J-equation and applications in Kähler geometry, stability, and cscK metrics.

## Contribution

It provides the first explicit formula for the optimal lower bound of Donaldson's J-functional and explores its implications in stability and existence of special metrics.

## Key findings

- Optimal lower bound formula explicitly computed for surfaces.
- The bound tends to minus infinity near the boundary of the Kähler cone.
- If the Lejmi-Székelyhidi conjecture holds, the bounds match algebraic stability criteria.

## 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 K\"ahler (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 K\"ahler cone, even when the underlying K\"ahler classes admit cscK metrics. As a final application we show that if the Lejmi-Sz\'ekelyhidi 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 K\"ahler cone, and there exists an optimal degeneration for uniform J-stability.