\(G_2\)-manifolds from \(K3\) surfaces with non-symplectic automorphisms
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.
scientific article; zbMATH DE number 6086949
scientific article