Non-symplectic Involution Research Articles

Non-symplectic involutions of normal K3 surfaces associated with Hirzebruch surfaces

For a non-symplectic involution g of a K3 surface X, the fixed point set of g and the quotient space X/⟨g⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X/\\langle g\\rangle $$\\end{document} are well studied. In this paper, we study a non-symplectic involution f of a normal K3 surface Y such that the smooth model of Y is a double cover K3 surface of a Hirzebruch surface and f is induced by its covering involution. We determine the fixed point set of f, the singular points of Y, and the quotient space Y/⟨f⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y/\\langle f\\rangle $$\\end{document}.

Quotient spaces of K3 surfaces by non-symplectic involutions fixing a curve of genus 8 or more

Abstract Let X be a K3 surface and let g be a non-symplectic involution of X such that the fixed points set contains a curve of genus 8 or more. In this paper, we show that the quotient space X/〈g〉 is determined by the fixed points set and the action of g on rational curves on X.

Heterotic strings on mathbbm{T} 3/ℤ2, Nikulin involutions and M-theory

We first describe the low energy dynamics of ten dimensional heterotic supergravity compactified on the smooth, flat 3-manifold mathbbm{T} 3/ℤ2, without supersymmetry, and explain how it arises from flat heterotic gauge fields. The semi-classical theory has both Coulomb and Higgs branches of non-supersymmetric vacua. We then give an exact worldsheet description as asymmetric orbifolds of mathbbm{T} 3, where the orbifold generator involves a Nikulin non-symplectic involution θ of the even self-dual lattice Γ(19,3). Along the way we briefly compare our findings with M-theory on K3/θ. Our construction gives a novel CFT description of the semi-classical field theory moduli space. In particular, the Wilson line parameters in the lattice I ⊂ Γ(19,3) of signature (19 − s, 1) which is invariant under θ, and in its orthogonal complement N , correspond respectively to Coulomb and Higgs branch moduli. There is a rich pattern of transitions amongst Higgs and Coulomb branches which we describe using the worldsheet theory.

K3 surfaces with a pair of commuting non-symplectic involutions

K3 surfaces with a pair of commuting non-symplectic involutions

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

On the non-symplectic involutions of the Hilbert square of a K3 surface

We investigate the interplay between the moduli spaces of ample -polarized IHS manifolds of type and of IHS manifolds of type with a non-symplectic involution with invariant lattice of rank one. In particular, we describe geometrically some new involutions of the Hilbert square of a K3 surface whose existence was proven in a previous paper of Boissière, Cattaneo, Nieper-Wisskirchen, and Sarti.

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

AbstractWe study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any . We show that is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any , there exist infinitely many admissible degrees for the polarization of the K3 surface S such that admits a non‐natural involution. This provides a generalization of the results of [7] for .

Counting associatives in compact G2 orbifolds

We describe a class of compact G2 orbifolds constructed from non-symplectic involutions of K3 surfaces. Within this class, we identify a model for which there are infinitely many associative submanifolds contributing to the effective superpotential of M-theory compactifications. Under a chain of dualities, these can be mapped to F-theory on a Calabi-Yau fourfold, and we find that they are dual to an example studied by Donagi, Grassi and Witten. Finally, we give two different descriptions of our main example and the associative submanifolds as a twisted connected sum.

Calabi–Yau Quotients of Hyperkähler Four-folds

AbstractThe aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.

On the nonsymplectic involutions of the Hilbert square of a K3 surface

Исследуется взаимосвязь между пространствами модулей обильно $\langle 2\rangle$-поляризованных НГС-многообразий типа $\mathrm{K3}^{[2]}$ и НГС-многообразий типа $\mathrm{K3}^{[2]}$ с антисимплектической инволюцией, инвариантная решетка которой имеет ранг один. В частности, дано геометрическое описание некоторых новых инволюций гильбертова квадрата K3-поверхности, существование которых было установлено в предыдущей работе Буассьера, Каттанео, Нипер-Висскирхена и Сарти. Библиография: 29 наименований.

On Chow groups of some hyperkahler fourfolds with a non-symplectic involution, II

This note is about hyperkahler fourfolds $X$ admitting a non-symplectic involution $\iota $. The Bloch-Beilinson conjectures predict the way $\iota $ should act on certain pieces of the Chow groups of $X$. The main result of this note is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $X/\iota $.

Verra Four-Folds, Twisted Sheaves, and the Last Involution

Abstract We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.

On the Chow groups of some hyperkähler fourfolds with a non-symplectic involution

This paper concerns hyperkähler fourfolds [Formula: see text] having a non-symplectic involution [Formula: see text]. The Bloch–Beilinson conjectures predict the way [Formula: see text] should act on certain pieces of the Chow groups of [Formula: see text]. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient [Formula: see text].

Non-symplectic involutions of irreducible symplectic manifolds of $$K3^{[n]}$$ K 3 [ n ] -type

This paper is concerned with non-symplectic involutions of irreducible symplectic manifolds of $K3^{[n]}$-type. We will give a criterion for deformation equivalence and use this to give a lattice-theoretic description of all deformation types. While moduli spaces of $K3^{[n]}$-type manifolds with non-symplectic involutions are not necessarily Hausdorff, we will construct quasi-projective moduli spaces for a certain well-behaved class of such pairs.

Lattice polarized irreducible holomorphic symplectic manifolds

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.

Rationality of the moduli spaces of 2-elementary 𝐾3 surfaces

K 3 K3 surfaces with non-symplectic involution are classified by open sets of seventy-five arithmetic quotients of type IV. We prove that those moduli spaces are rational except two classical cases.

The Berglund–Hübsch–Chiodo–Ruan mirror symmetry for K3 surfaces

We prove that the mirror symmetry of Berglund, Hübsch, Chiodo and Ruan, applied to K3 surfaces with a non-symplectic involution, coincides with the lattice mirror symmetry.

Formula omitted]-manifolds from [formula omitted] surfaces with non-symplectic automorphisms

We show that K3 surfaces with non-symplectic automorphisms of prime order can be used to construct new compact irreducible G2-manifolds. This technique was carried out in detail by Kovalev and Lee for non-symplectic involutions. We use the Chen–Ruan orbifold cohomology to determine the Hodge diamonds of certain complex threefolds, which are the building blocks for this approach.

The unirationality of the moduli spaces of 2-elementary K 3 surfaces

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product, we describe configuration spaces of 5 ⩽d⩽ 8 points in ℙ2 as arithmetic quotients of type IV.