\(G_2\)-manifolds from \(K3\) surfaces with non-symplectic automorphisms
type: Item
instance of: scholarly article
title: \(G_2\)-manifolds from \(K3\) surfaces with non-symplectic automorphisms
zbMATH Open document ID: 1267.53049
author: Max Pumperla
author: Frank Reidegeld
published in: Journal of Geometry and Physics
publication date: 2012-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.2623
review text: The authors construct new examples of compact irreducible \(G_2\)-manifolds from \(K3\) surfaces with non-symplectic automorphisms of prime order. They follow the approach by Kovalev and Lee. In Section 6, starting from a \(K3\) surface with a non-symplectic automorphism of prime order \(p\), it is shown how to construct threefolds of non-symplectic type \(\overline W\) and \(W = \overline W \setminus D\), such that \((\overline W, D)\) is an admissible pair (Definition 2.3). Hodge numbers are computed by using the Chen-Ruan orbifold cohomology. In Section 7, the authors glue admissible pairs \((\overline W_j, D_j)\), \(j = 1, 2\), satisfying the matching condition (Definition 2.4) in order to get \(G_2\)-manifolds. They restrict themselves to the case where \((\overline W_1, D_1)\) is of non-symplectic type with \(p = 3\) and \((\overline W_2, D_2)\) is of Fano-type. They compute the Betti numbers of the resulting \(G_2\)-manifold \(M\). Finally, they give some explicit examples where \(b^2(M) = 42\) and \(b^3(M) = 91, 113\) and thus, different from known examples.
reviewed by: Silvio Reggiani
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6086949
zbMATH Keywords: \(G_2\)-manifolds
zbMATH Keywords: K3 surfaces
zbMATH Keywords: generalized connected sums
MaRDI profile type: Publication
OpenAlex ID: W1989904372
arXiv ID: 1110.2623
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DOI: 10.1016/J.GEOMPHYS.2012.06.002
description: scientific article; zbMATH DE number 6086949