symplectic - Annals of Mathematics 169(2009 269–313...

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Unformatted text preview: Annals of Mathematics , 169 (2009), 269–313 Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic By Igor V. Dolgachev and JongHae Keum * Abstract We show that Mukai’s classification of finite groups which may act sym- plectically 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. 1. Introduction A remarkable work of S. Mukai [Mu] gives a classification of finite groups which can act on a complex algebraic K3 surface X leaving invariant its holo- morphic 2-form (symplectic automorphism groups). Any such group turns out to be isomorphic to a subgroup of the Mathieu group M 23 which has at least five orbits in its natural action on a set of 24 elements. A list of maximal sub- groups with this property consists of 11 groups, each of which can be realized on an explicitly given K3 surface. A different proof of Mukai’s result was given later by S. Kond¯ o [Ko]. G. Xiao [Xiao] classified all possible topological types of a symplectic action. Neither Mukai’s nor Kond¯ o’s proof extends to the case of K3 surfaces over algebraically closed fields of positive characteristic p . In fact there are known examples of surfaces over a field of positive character- istic whose automorphism group contains a finite symplectic subgroup which is not realized as a subgroup of M 23 (e.g. the Fermat quartic over a field of characteristic 3, or the surface from [DKo] over a field of characteristic 2). *Research of the first author is partially supported by NSF grant DMS-0245203. Research of the second named author is supported by KOSEF grant R01-2003-000-11634-0. 270 IGOR V. DOLGACHEV AND JONGHAE KEUM The main tool used in Mukai’s proof is the characterization of the represen- tation of a symplectic group G ⊂ Aut( X ) on the 24-dimensional cohomology space H * ( X, Q ) = H ( X, Q ) ⊕ H 2 ( X, Q ) ⊕ H 4 ( X, Q ) . Using the Lefschetz fixed-point formula and the description of possible finite cyclic subgroups of Aut( X ) and their fixed-point sets due to Nikulin [Ni1], one can compute the value χ ( g ) of the character of this representation at any element of finite order n . It turns out that χ ( g ) = ε ( n ) for some function ε ( n ) and the same function describes the character of the 24-permutation representation of M 23 . A representation of a finite group G in a finite-dimensional vector space of dimension 24 over a field of characteristic 0 is called a Mathieu representation if its character is given by...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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symplectic - Annals of Mathematics 169(2009 269–313...

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