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Oberseminar WiSe 2009/10 Zusammenfassung des Vortrags von A. Garbagnati (Milano, ITA)

Elliptic K3 surfaces with abelian and dihedral groups of symplectic automorphisms

A group G acts symplectically on a K3 surface X if it leaves invariant the 2-holomorphic form of the K3. In '79, Nikulin classified the finite abelian groups acting symplectically on K3 surfaces, and proved that the isometries induced by G on H 2( X, Z) are essentially unique, i.e. they do not depend on X. Later, Nikulin's results on finite abelian groups of symplectic automorphisms were generalized to the non abelian groups (by Mukai, Xiao, Kondo, Whitcher).

The uniqueness of the isometries induced by G on H 2( X, Z) suggests to construct particular examples of K3 surfaces with symplectic automorphisms and to deduce by these examples general results. During this talk I construct K3 surfaces X admitting abelian and dihedral groups G of symplectic automorphisms, using elliptic fibration on K3 surfaces, and I describe the desingularization of the quotient X/G (which is again an elliptic K3 surface). I prove that if a K3 surface (not necessarily elliptic) admits Z/5Z as group of symplectic automorphisms, then it admits also a symplectic involution, and actually the dihedral group of order 10 as group of symplectic automorphisms. Moreover I compute certain lattices, related to the presence of symplectic automorphisms on a K3 surface, and, as a consequence, I prove that lattices which are already known in literature are isomorphic and I describe the Mordell-Weil lattice of an ellliptic K3 surface described by Kloosterman as example of an elliptic K3 surface with Mordell-Weil rank equal to 15.