Symmetric symplectic homotopy K3 surfaces
نویسنده
چکیده
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.
منابع مشابه
The Modularity of K3 Surfaces with Non-symplectic Group Actions
We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show tha...
متن کاملSymplectic Rational Blowdowns
We prove that the rational blowdown, a surgery on smooth 4-manifolds introduced by Fintushel and Stern, can be performed in the symplectic category. As a consequence, interesting families of smooth 4-manifolds, including the exotic K3 surfaces of Gompf and Mrowka, admit symplectic structures. A basic problem in symplectic topology is to understand what smooth manifolds admit a symplectic struct...
متن کاملRigidity of Periodic Diffeomorphisms on Homotopy K3 Surfaces
In this paper we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. This gives a negative answer for period 3 to Problem 4.124 in the Kirby’s problem list. In addition, we give an obstruction in terms of the rationality and sign of the spin numbers to the existence of a periodic diffeomorphism of odd prime order acting triv...
متن کاملSymplectic Automorphisms and the Picard Group of a K3 Surface
Let X be a K3 surface, and let G be a finite group acting on X by automorphisms. The action of G on X induces an action on the cohomology of X . We assume G acts symplectically: that is, G acts as the identity on H(X). In this case, the minimum resolution Y of the quotient X/G is itself a K3 surface. Nikulin classified the finite abelian groups which act symplectically on K3 surfaces by analyzi...
متن کاملOn Symplectic and Non–symplectic Automorphisms of K3 Surfaces
In this paper we investigate when the generic member of a family of complex K3 surfaces admitting a non–symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain c...
متن کامل