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アーカイブ 研究アーカイブ KIASとの学術交流協定締結記念講演会
理1号館 509講義室 名古屋大学大学院多元数理科学研究科
“Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic”
A remarkable work of S. Mukai gives a classification of finite groups which can act on a complex algebraic K3 surface leaving invariant its holomorphic 2-form (symplectic automorphism groups). Any such group turns out to be isomorphic to a subgroup of the Mathieu group M23 which has at least 5 orbits in its natural action on the set of 24 elements. A list of maximal subgroups with this property consists of 11 groups, each of these can be realized on an explicitly given K3 surface. The subsequent proofs of Mukai's result were given by different methods by S. Kondo and Xiao. None of these proofs extends to the case of K3 surfaces over algebraically closed fields of positive characteristic p.
In this talk we outline a joint work with I. Dolgachev, namely that the ideas of Mukai's proof can be extended to positive characteristic p > 11, and consequently that we have the same list of groups for p > 11.
For p ≤ 11 there are known examples of K3 surfaces over a field of characteristic p whose automorphism group contains a finite symplectic subgroup which is not realized as a subgroup of M23.