Optimal lower bounds for Donaldson's J-functional.

TY - JOUR

T1 - Optimal lower bounds for Donaldson's J-functional.

AU - Sjöström Dyrefelt, Zakarias

PY - 2020/11/18

Y1 - 2020/11/18

N2 - 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.

AB - 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.

KW - Calabi dream manifolds

KW - Donaldson's J-functional

KW - J-equation

KW - J-stability

KW - Properness

KW - Tian's alpha invariant

KW - cscK metrics

UR - http://www.scopus.com/inward/record.url?scp=85091894819&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2020.107271

DO - 10.1016/j.aim.2020.107271

M3 - Journal article

SN - 0001-8708

VL - 374

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107271

ER -