We give a classification of all non-symplectic automorphisms of prime order p acting on irreducible holomorphic symplectic fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p = 2, 3 and 7 ≤ p ≤ 19. Our classification relates the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients with some invariants of the fixed locus and we provide explicit examples. As an application, we find new examples of non-natural non-symplectic automorphisms.
The main goal of this paper is the study of fixed points of a symplectic involution over an irreducible holomorphic symplectic manifold of dimension 4 such that b2=23. We show that there are only three possibilities for the number of fixed points and of fixed K3 surfaces. We conjecture that only one case can actually occur, the one with 28 isolated fixed points and 1 fixed K3 surface, and that such an involution can never fix an abelian surface. We provide evidence for the conjecture by verifying it in some examples, as the Hilbert scheme of a K3 surface, the Fano variety of a cubic in ℙ5 and the double cover of an Eisenbud–Popescu–Walter sextic.
Abstract. We prove that there exists a holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of two points on a K3 surface that admits a non-symplectic automorphism of order 23, that is the maximal possible prime order in this deformation family. The proof uses the theory of ideal lattices in cyclomotic fields.
Let X be a smooth projective surface over C and let L be a line bundle on X generated by its global sections. Let φ L : X −→ P r be the morphism associated to L; we investigate the μ−stability of φ * L T P r with respect to L when X is either a regular surface with p g = 0, a K3 surface or an abelian surface. In particular, we show that φ * L T P r is μ− stable when X is K3 and L is ample and when X is abelian and L 2 ≥ 14.
We generalize Nikulin's and Dolgachev's lattice-theoretical mirror symmetry
for K3 surfaces to lattice polarized higher dimensional irreducible holomorphic
symplectic manifolds. In the case of fourfolds of $K3^{\left[2\right]}-$type we
then describe mirror families of polarized fourfolds and we give an example
with mirror non-symplectic involutions.Comment: Comments are welcom
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