Mathematisches Forschungsinstitut Oberwolfach

Report No. 39/2010
DOI: 10.4171/OWR/2010/39

Organised by
Jean-Pierre Demailly, Grenoble
Klaus Hulek, HannoverThomas Peternell, Bayreuth

August 29th – September 4th, 2010

Abstracts
Enriques surfaces with many (semi-)symplectic automorphisms
Shigeru Mukai
An automorphism of a K3 surface S is sympletic if it acts on H 0 (OS (KS )) trivially. All ﬁnite groups which have symplectic actions on K3 surfaces are classiﬁedin terms of the Mathieu group M24 by Mukai [4] and Kondo [2]. An automorphismof an Enriques surface S is semi-symplectic if it acts on H 0 (OS (2KS )) trivially. A
smart classiﬁcation similar to K3 surfaces is desirable for semi-symplectic actionsof Enriques surfaces but still far from complete investigation. Here I propose a
restricted class of semi-symplectic actions.
Deﬁnition An eﬀective semi-symplectic action of a ﬁnite group G on an Enriques surface is M -sympletic if the Lefschetz number of g equals 4 for every
automorphism g ∈ G of order 2 and 4.
Here the Lefschetz number of an automorphism σ is the Euler number of theﬁxed point locus Fix σ, and equal to the trace of the cohomology action of σ onH ∗ (S, Q).
M -semi-symplectic actions are closely related to the symmetric group S6 of degree 6 via the Mathieu group M12 though S6 itself has no semi-symplectic actions.
It is known that S6 has six maximal subgroups upto conjugacy, and four modulo
automorphisms. The four subgroups are

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Oberwolfach Report 39(1) the alternating group A6 ,(2) the symmetric group S5 of degree 5,(3) (C3 )2 .D8 , the normalizer of a 3-Sylow subgroup, and(4) the direct product S4 × C2 ,where Cn and Dn denote a cyclic and a dihedral group of order n, respectively.
Theorem The three maximal subgroups A6 , S5 , (C3 )2 .D8 and the abelian group(C2 )3 have M -semi-symplectic actions on Enriques surfaces.
Remark By Kondo [1], there are two Enriques surfaces whose automorphismgroups are isomorphic to S5 . One is called type VII and the other is the quotientof the Hessian of a special cubic surface (type VI). The action of S5 is M -semisymplectic for the former and not for the latter.
The action of the three maximal subgroups are constructed reﬁning the methodof [5]. We use(1) embeddings of S6 into the Mathieu group M12 ,(2) the action of M12 × C2 on the Leech lattice, and(3) Torelli type theorem for Enriques surfaces.
An Enriques surface S = Km(E1 ×E2 )/ε of Lieberman type has a semi-symplectic
action of (C2 )4 by translation by 2-torsion points. One involution σ ∈ (C2 )4 isnumerically trivial in the sense of [3], that is, its Lefschetz number is the maximal(= 12). Moreover, the action of (C2 )4 is M -semi-symplectic except for σ. Hence
S has an M -semi-symplectic action of (C2 )3Question Is a ﬁnite group isomorphic to a proper subgroup of the symmetricgroup S6 , if it has an (eﬀective) M -semi-symplectic action on an Enriques surface?
The deﬁnition of M -semi-symplectic action is modeled on the permutationgroup M12 of degree 12. The permutation type of g ∈ M12 depends only onits order n if it has a ﬁxed point (on the operator domain of cardinality 12). Thetype and the number of ﬁxed points µ+ (n) are as follows.
n1permutation type (1)µ+ (n)
122(2)443(3)334(4)2
45(5)226(6)(3)(2)1

811(8)(2) (11)21

It is well known that a symplectic involution of a K3 surface have exactly 8 ﬁxedpoints. But for an involution σ of an Enriques surface, the ﬁxed point set Fix σ isnot necessarily ﬁnite and the Lefschetz number varies from −4 to 12. (Note thatevery involution of an Enriques surface is semi-symplectic.) The required number
4 in our deﬁnition is one half of 8, the mean of −4 and 12 and equal to µ+ (2).
A semi-symplectic action of G on an Enriques surface is M -semi-symplectic if
and only if the Lefschetz number and µ+ are the same on G since the order ofsemi-symplectic automorphism is either ≤ 6 or ∞ by H. Ohashi.

3References[1] Kondo, S.: Enriques surfaces with ﬁnite automorphism groups, Japan. J. Math., 12(1986),
191–282.
[2] Kondo, S.: Niemeier lattices, Mathieu groups and ﬁnite groups of symplectic automorphismsof K3 surfaces, Duke Math. J. 92(1998), 593–598.
[3] Mukai, S. and Namikawa, Y.: Automorphisms of Enriques surfaces which act trivially on thecohomology groups, Invent. math., 77(1984), 383–397.
[4] Mukai, S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent.
math., 94(1988), 183–221.
[5] Mukai, S.: Lattice theoretic construction of symplectic actions on K3 surfaces, Appendix to[2], Duke Math. J. 92(1998), 599–603.

Reporter: Arthur Prendergast-Smith