K3 surfaces with non-symplectic involution and compact irreducibleG2-manifolds

Abstract

AbstractWe consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomyG2developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonicalK3 divisors and the latterK3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.In this paper, we give a large new class of suitable algebraic threefolds using theory ofK3 surfaces with non-symplectic involution due to Nikulin. These threefolds arenotobtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducibleG2-manifolds. ‘Geography’ of the values of Betti numbersb2,b3for the new (and previously known) examples of irreducibleG2manifolds is also discussed.