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K3 surfaces with non-symplectic involution and compact irreducible G2-manifolds

Published online by Cambridge University Press:  10 June 2011

ALEXEI KOVALEV  and
NAM-HOON LEE
ALEXEI KOVALEV
Affiliation:
DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB. e-mail: [email protected]
NAM-HOON LEE
Affiliation:
Department of Mathematics Education, Hongik University 42-1, Sangsu-Dong, Mapo-Gu, Seoul 121-791, Korea. e-mail: [email protected]
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Abstract

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G2 developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors and the latter K3 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 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 G2-manifolds. ‘Geography’ of the values of Betti numbers b2, b3 for the new (and previously known) examples of irreducible G2 manifolds is also discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 193 - 218
Copyright
Copyright © Cambridge Philosophical Society 2011

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