TY - JOUR
T1 - K-stability of cubic threefolds
AU - Liu, Yuchen
AU - Xu, Chenyang
N1 - Funding Information:
Liu’s work was partially supported by National Science Foundation grant DMS-1362960. Xu’s work was partially supported by The Chinese National Science Fund for Distinguished Young Scholars (11425101).
Publisher Copyright:
© 2019.
PY - 2019
Y1 - 2019
N2 - We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension 3 of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of 3-dimensional canonical and terminal singularities, which was established during the study of the explicit 3-dimensional minimal model program.
AB - We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension 3 of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of 3-dimensional canonical and terminal singularities, which was established during the study of the explicit 3-dimensional minimal model program.
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U2 - 10.1215/00127094-2019-0006
DO - 10.1215/00127094-2019-0006
M3 - Article
AN - SCOPUS:85073028211
SN - 0012-7094
VL - 168
SP - 2029
EP - 2073
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 11
ER -