On K-polystability of cscK manifolds with transcendental cohomology class: with an appendix by R. Dervan

TY - JOUR

T1 - On K-polystability of cscK manifolds with transcendental cohomology class

T2 - with an appendix by R. Dervan

AU - Sjöström Dyrefelt, Zakarias

PY - 2020

Y1 - 2020

N2 - In this paper we study K-polystability of arbitrary (possibly non-projective) compact Kähler manifolds admitting holomorphic vector fields. As a main result we show that existence of a constant scalar curvature Kähler (cscK) metric implies geodesic K-polystability, in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen–Tang [28] we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds, so our main result recovers a possibly stronger version of results of Berman–Darvas–Lu [10] in this case. As a key part of the proof we also study subgeodesic rays with singularity type prescribed by singular test configurations and prove a result on asymptotics of the K-energy functional along such rays. In an appendix by R. Dervan it is moreover deduced that geodesic K-polystability implies equivariant K-polystability. This improves upon the results of [39] and proves that existence of a cscK (or extremal) Kähler metric implies equivariant K-polystability (resp. relative K-stability).

AB - In this paper we study K-polystability of arbitrary (possibly non-projective) compact Kähler manifolds admitting holomorphic vector fields. As a main result we show that existence of a constant scalar curvature Kähler (cscK) metric implies geodesic K-polystability, in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen–Tang [28] we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds, so our main result recovers a possibly stronger version of results of Berman–Darvas–Lu [10] in this case. As a key part of the proof we also study subgeodesic rays with singularity type prescribed by singular test configurations and prove a result on asymptotics of the K-energy functional along such rays. In an appendix by R. Dervan it is moreover deduced that geodesic K-polystability implies equivariant K-polystability. This improves upon the results of [39] and proves that existence of a cscK (or extremal) Kähler metric implies equivariant K-polystability (resp. relative K-stability).

UR - http://www.scopus.com/inward/record.url?scp=85068154932&partnerID=8YFLogxK

U2 - 10.1093/imrn/rny094

DO - 10.1093/imrn/rny094

M3 - Journal article

SN - 1073-7928

VL - 2020

SP - 2769

EP - 2817

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 9

ER -