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We show that Mukai’s classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under assumptions that (i) the order of the group is coprime to $p$ and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without assumption (ii) we classify all possible new groups which may appear. We prove that assumption (i) on the order of the group is always satisfied if $p > 11$. For $p=2,3,5,11$, we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by $p$ which are not contained in Mukai’s list.
@article{10_4007_annals_2009_169_269,
author = {Igor Dolgachev and JongHae Keum},
title = {Finite groups of symplectic automorphisms of {K3} surfaces in positive characteristic},
journal = {Annals of mathematics},
pages = {269--313},
publisher = {mathdoc},
volume = {169},
number = {1},
year = {2009},
doi = {10.4007/annals.2009.169.269},
mrnumber = {2480606},
zbl = {1187.14047},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.269/}
}
TY - JOUR AU - Igor Dolgachev AU - JongHae Keum TI - Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic JO - Annals of mathematics PY - 2009 SP - 269 EP - 313 VL - 169 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.269/ DO - 10.4007/annals.2009.169.269 LA - en ID - 10_4007_annals_2009_169_269 ER -
%0 Journal Article %A Igor Dolgachev %A JongHae Keum %T Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic %J Annals of mathematics %D 2009 %P 269-313 %V 169 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.269/ %R 10.4007/annals.2009.169.269 %G en %F 10_4007_annals_2009_169_269
Igor Dolgachev; JongHae Keum. Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic. Annals of mathematics, Tome 169 (2009) no. 1, pp. 269-313. doi: 10.4007/annals.2009.169.269
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