Isometries of Ideal Lattices and Hyperkähler Manifolds

Abstract

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.

Complex ball quotients from manifolds of K3[]-type

“…We recall now some of the results of [12] which contains the classification of nonsymplectic automorphisms of prime order 3 ≤ p ≤ 19, p = 5 and partial results on involutions, completing the ones in [10]. The cases p = 5 and p = 23 were then discussed respectively in [29] and in [11]. Such automorphisms are classified in terms of their invariant sublattice T (see [12, Appendix A] and [29, Section 3.4]).…”

“…Our classification relates certain invariants of the fixed locus with the isometry classes of two natural lattices, associated to the action of the automorphism on the second integral cohomology group. Then, in [11] and in [29] the cases p = 23 and p = 5 were solved, thus completing the classification for IHS − K3 [2] and prime order.…”

Non-symplectic automorphisms of odd prime order on manifolds of $$K3^{\left[ n\right] }$$-type

“…We are interested in the following triples (p, m, a): (3,9,5) and (3,8,6) for n = 3; (3, 10, 3), (3,9,4), (3,8,5) for n = 4. For each of these cases, let T, S be the corresponding lattices in Table 1…”

“…Consider now the case where (p, m, a) is one of the admissible triples (3,9,5), (3,8,6) for n = 3. In this case Pic(Σ) ∼ = U (3) ⊕ A 2 ⊕ M ′ , therefore -if {e 1 , e 2 } is a basis for U (3) and {δ 1 , δ 2 } a basis for A 2 -we can take the primitive element of square fourH = e 1 + e 2 + δ 1 ∈ Pic(Σ).…”

“…As we observed in §3.3, there are no admissible triples with m = 11. • The triple(3,9,5) is now admissible: here S = U (3) ⊕2 ⊕ E 6 ⊕ E 8 ,while sign(T ) =(1,4) andq T = −q S ⊕ q L ∼ = Z 3Z 4The existence of a lattice T with these invariants is proved applying [43, Theorem 1.10.1] and there is a unique isometry class in the genus of T by[22, Chapter 15, Theorem 21]. In particular, we can take T = U (3) ⊕ Ω, where Ω is the even lattice of rank three whose bilinear form is defined by the matrixΩ We have sign(Ω) = (0, 3) and q Ω = Z q U(3)⊕Ω ∼ = −q S ⊕ q L (using [43, Proposition 1.8.2]).…”