Existence of cscK metrics on smooth minimal models

TY - JOUR

T1 - Existence of cscK metrics on smooth minimal models

AU - Sjöström Dyrefelt, Zakarias

PY - 2022/3

Y1 - 2022/3

N2 - Given a compact Kähler manifold X it is interesting to ask whether it admits a constant scalar curvature Kähler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kähler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song and extends their main result from K_X semi-ample to K_X nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component Aut_0(X) of a compact Kähler manifold with K_X nef is either trivial or a complex torus.

AB - Given a compact Kähler manifold X it is interesting to ask whether it admits a constant scalar curvature Kähler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kähler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song and extends their main result from K_X semi-ample to K_X nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component Aut_0(X) of a compact Kähler manifold with K_X nef is either trivial or a complex torus.

KW - cscK metrics

KW - Properness of energy functionals

KW - Minimal Model Program

UR - http://www.scopus.com/inward/record.url?scp=85136180832&partnerID=8YFLogxK

U2 - 10.2422/2036-2145.202005_021

DO - 10.2422/2036-2145.202005_021

M3 - Journal article

SN - 0391-173X

VL - XXIII

SP - 223

EP - 232

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

JF - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

IS - 1

ER -