No abstract

We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.ICHIRO SHIMADA AND DAVIDE CESARE VENIANI but X 7 does. Hence, following Vinberg, we call the Enriques surfaces covered by X 7 the most algebraic Enriques surfaces.Theorem 1.0.1. The singular K3 surface X 7 of discriminant 7 has exactly two Enriques involutionsε I andε II up to conjugation in Aut(X 7 ). Let Y I and Y II be the quotient Enriques surfaces corresponding toε I andε II , respectively. Then Aut(Y I ) is finite of order 8, and Aut(Y II ) is finite of order 24.Nikulin [26] and Kondo [19] classified all complex Enriques surfaces whose automorphism group is finite. It turns out that these Enriques surfaces are divided into 7 classes I, II, . . . , VII, which we call Nikulin-Kondo type. See Kondo [19] for the properties of these Enriques surfaces. Corollary 1.0.2. The most algebraic Enriques surfaces have finite automorphism groups and their Nikulin-Kondo types are I and II.In Section 6 of this paper, we give explicit models of the most algebraic Enriques surfaces Y I and Y II as Enriques sextic surfaces.Remark 1.0.3. The Néron-Severi lattice and the automorphism group of X 7 were determined by Ujikawa [39]. Elliptic fibrations on X 7 were studied by Harrache-Lecacheux [12] and Lecacheux [21].Remark 1.0.4. Mukai [23] also realized that X 7 has Enriques involutions that produce Enriques surfaces of Nikulin-Kondo type I and II.Ohashi [27] gave a lattice theoretic method to enumerate Enriques involutions on certain K3 surfaces. He then classified in [28] all Enriques involutions on the Kummer surface Km(Jac(C)) associated with the jacobian variety of a generic curve C of genus 2. We refine and generalize Ohashi's method. Our main result, namely Theorem 3.1.9, applies to any K3 surface, and we use it in the case of singular K3 surfaces to compile Table 3.1.For some K3 surfaces X, the group Aut(X) can be calculated by Borcherds method ([4], [5]); for instance, Kondo [20] implemented it in order to compute Aut(Km(Jac(C))). We apply Borcherds method in order to calculate the automorphism group of some of singular K3 surfaces X, and to write the action of Aut(X) on the nef chamber of X explicitly. Building on this data, we enumerate all Enriques involutions up to conjugation, and, using also a result of the preprint [7] (see Section 2.9), we calculate the automorphism group of the Enriques surfaces covered by these K3 surfaces.Note that the enumeration of Enriques involutions by Ohashi's method and by Borcherds method are carried out independently. The results are, of course, consistent. We hope that these methods will be applied to many other K3 surface...

We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-Hübsch duality. Our method is a variant of the so-called Landau-Ginzburg/Calabi-Yau correspondence of Calabi-Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi-Yau category and feature hypersurfaces of general type.The above theorem provides many examples of Calabi-Yau mirror pairs unknown before. These statements turn into ordinary cohomology statement whenever crepant resolutions on the two sides exist.Crepant resolution conjecture [28]. Finally, very recently, Hull, Israel and Sarti used mirror symmetry for K3 surfaces to form "non-geometric backgrounds" in the physics paper [21].1.10. Contents. In §2 we recall terminology briefly. In §3 we recall some basic definitions about Berglund-Hübsch invertible polynomials. In §4 we treat orbifold cohomology, its σorbifold variant, and we prove the compatibility result (5) stated above. In §5 we prove all the relevant statements at the level of Landau-Ginzburg state spaces. In §6 we derive the corresponding geometric versions stated above, see in particular §6.3 with some examples. Relation to K3 surfaces is treated in §6.4; we compare to the approach of [2] in Example 6.4.3.Higher dimensional Borcea-Voisin mirror theorem is deduced in §6.5.Aknowledgements. We are grateful to Alfio Ragusa, Francesco Russo and Giuseppe Zappalà for organising Pragmatic 2015, where this work started. We are grateful to London 2. Terminology 2.1. Conventions. We work with schemes and stacks over the complex numbers. All schemes are Noetherian and separated. By linear algebraic group we mean a closed subgroup of GL m (C) for some m. We often use strict Henselizations in order to describe a stack or a morphism between stacks locally at a closed point: by "local picture of X at the geometric point x ∈ X" we mean the strict Henselization of X at x.2.2. Notation. We list here notation that occurs throughout the entire paper.V K the invariant subspace of a vector space V linearized by a finite group K; P(w w w) the quotient stack [(C n \ 0 0 0)/G m ] with w w w-weighted G m -action; Z(f ) the variety defined as zero locus of f ∈ C[x 1 , . . . , x n ].Remark 2.2.1 (zero loci). We add the subscript P(w w w) when we refer to the zero locus in P(w w w) of a polynomial f which is w w w-weighted homogeneous. In this way we haveRemark 2.2.2 (graphs and maps). Given an automorphism α of X we write Γ α for the graph X → X × X. However, to simplify formulae, we often abuse notation and use α for the graph Γ α as well as the automorphism. In this way, the diagonal ∆ : X → X × X will be often written as id X or simply id. Berglund-Hübsch polynomialsThe setup presented here is due to Berglund-Hübsch [3]. We also refer to [4,15, 16,24,22]. It can be motivated as the simplest generalization of Gree...

We classify the transcendental lattices of all K3 surfaces covering an Enriques surface. In order to do so, we classify all genera of positive definite lattices that only contain lattices with a vector of square 1.

In this paper we consider irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady, and their symplectic birational transformations of finite order. We study the induced isometries on the Beauville-Bogomolov-Fujiki lattice, classifying all possible invariant and coinvariant sublattices, and providing explicit examples. As a consequence, we show that the isometry induced by a symplectic automorphism of finite order is necessarily trivial. This paper is a first step towards the classification of all finite groups of symplectic birational transformations.