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

By A. Kovalev, Nam-Hoon Lee. Mathematical Proceedings of the Cambridge Philosophical Society, 2008

72 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c8f2ca6f083efe347d38a1f4f777551e2c6fb50d

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Citations in the corpus

G2-manifolds and associative submanifolds via semi-Fano 3-folds (2012, methodology): “Another kind of building blocks was defined by Kovalev and Lee [45] from K3 surfaces S with non-symplectic involution, ie with an involution τ acting as −1 on H2,0(S).”
Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds (2012, methodology): “ACyl Calabi–Yau 3–folds play a distinguished role because they can also be used as building blocks in Kovalev’s twisted connected sum construction of compact manifolds with holonomy G2 : see Kovalev [57], Kovalev–Lee [58] and the more recent developments in Corti–Haskins–Nordström–Pacini [17].”
Notes on the octonions (2010, methodology): “There are however a number of construction techniques (all based on gluing methods): Joyce’s generalized Kummer construction for G2– and Spin(7)–manifolds [20, 21, 18, 19] based on resolving orbifolds of the form T7/Γ and T8/Γ; a method of Joyce’s for constructing Spin(7)–manifolds from real singula”
Asymptotically cylindrical Calabi-Yau manifolds (2012, methodology): “By modifying the construction of [26], we can find admissible orbifolds M with m > 1, as follows.”
G2-instantons over twisted connected sums (2013, methodology): “Kovalev and Lee [16] construct building blocks starting from K3 surfaces with non-symplectic involutions, by taking the product with P1, dividing by Z2 and blowing up the resulting singularities.”