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

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

2008-10-06

arxiv.org/abs/0810.0957v4

Math. Proc. Cambridge Philos. Soc. 151 (2011), 193-218

Mathematics

Differential Geometry

30 pages; v2: misprint in the abstract corrected (no changes in the paper); v3: formula for b^3(M) in Theorem 2.5 corrected, r

Scientific paper

10.1017/S030500411100003X

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.

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