K3 surfaces with non-symplectic involution and compact irreducible G_2-manifolds

TL;DR

This paper introduces new algebraic threefolds derived from K3 surfaces with non-symplectic involution, enabling the construction of novel compact irreducible G_2-manifolds via the connected-sum method, expanding the known examples and their topological invariants.

Contribution

It provides new algebraic threefolds not obtainable from Fano threefolds, using Nikulin's theory, and constructs previously unknown G_2-manifolds with specific Betti number configurations.

Findings

New examples of compact irreducible G_2-manifolds constructed.

Betti number 'geography' of these manifolds analyzed.

Extension of the connected-sum construction method.

Abstract

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds were previously obtained in math.DG/0012189 by blowing up curves in Fano threefolds. In this paper, we give further suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds, as above, and admit matching pairs leading to topologically new examples of compact irreducible G_2-manifolds. `Geography' of the values of Betti numbers b^2,b^3 for the new (and previously known) examples of compact irreducible G_2 manifolds is also discussed.