TY - JOUR

T1 - The Nielsen realization problem for K3 surfaces

AU - Farb, Benson

AU - Looijenga, Eduard

N1 - Publisher Copyright:
© 2024 International Press, Inc.. All rights reserved.

PY - 2024/6

Y1 - 2024/6

N2 - The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces M asks: when can a finite group G of mapping classes of M be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version for involutions. Unlike the case of 2-manifolds, some G are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant LG that determines in many cases whether G is realizable or not, and apply this invariant to construct an S4 action by isometries of some Ricci-flat metric on M that preserves no complex structure. We also show that the subgroups of Diff(M) of a given prime order p which fix pointwise some positive-definite 3-plane in H2(M; R) and preserve some complex structure on M form a single conjugacy class in Diff(M) (it is known that then p ∈ {2, 3, 5, 7}).

AB - The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces M asks: when can a finite group G of mapping classes of M be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version for involutions. Unlike the case of 2-manifolds, some G are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant LG that determines in many cases whether G is realizable or not, and apply this invariant to construct an S4 action by isometries of some Ricci-flat metric on M that preserves no complex structure. We also show that the subgroups of Diff(M) of a given prime order p which fix pointwise some positive-definite 3-plane in H2(M; R) and preserve some complex structure on M form a single conjugacy class in Diff(M) (it is known that then p ∈ {2, 3, 5, 7}).

UR - http://www.scopus.com/inward/record.url?scp=85196353970&partnerID=8YFLogxK

U2 - 10.4310/jdg/1717772420

DO - 10.4310/jdg/1717772420

M3 - Article

AN - SCOPUS:85196353970

SN - 0022-040X

VL - 127

SP - 505

EP - 549

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

IS - 2

ER -