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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: a.kovalev@dpmms.cam.ac.uk
NAM-HOON LEE
Affiliation:
Department of Mathematics Education, Hongik University 42-1, Sangsu-Dong, Mapo-Gu, Seoul 121-791, Korea. e-mail: nhlee@hongik.ac.kr
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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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References

REFERENCES

[1]Alexeev, V. and Nikulin, V. V.. Del Pezzo and K3 surfaces. MSJ Mem. 15 (2006).Google Scholar
[2]Barth, W., Hulek, K., Peters, C. and van de Ven, A.. Compact Complex Surfaces (Springer-Verlag, 2004).CrossRefGoogle Scholar
[3]Borcea, C.. K3 surfaces with involution and mirror pairs of Calabi–Yau manifolds. In Mirror symmetry, II, AMS/IP Stud. Adv. Math. 1 (1997), pp. 717–743.Google Scholar
[4]Corti, A., Haskins, M., Nordström, J. and Pacini, T.. G 2 manifolds and associative submanifolds via weak Fano 3-folds, in preparation.Google Scholar
[5]Dolgachev, I.. Integral quadratic forms: applications to algebraic geometry (after V. Nikulin). In Bourbaki seminar Vol. 1982/83. Astérisque 105–106 (1983), pp. 251–278.Google Scholar
[6]Dolgachev, I. V.. Mirror symmetry for lattice polarized K3 surfaces. Algebraic geometry, 4. J. Math. Sci. 81 (1996), 2599–2630.CrossRefGoogle Scholar
[7]Griffiths, P. and Harris, J.. Principles of Algebraic Geometry (John Wiley & Sons, 1978).Google Scholar
[8]Iskovskih, V. A.. Fano threefolds. I. II. Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 516–562, 717, and 42 (1978), 506–549. English translation: Math. USSR Izvestia 11 (1977), 485–527 and 12 (1978) 469–506.Google Scholar
[9]Iskovskikh, V. A. and Prokhorov, Yu. G.. Fano varieties. Algebraic geometry, V. Encyclopaedia of Math. Sciences, vol. 47 (Springer, 1999).Google Scholar
[10]Joyce, D.. Compact Manifolds with Special Holonomy (Oxford University Press, 2000).CrossRefGoogle Scholar
[11]Joyce, D.. Riemannian Holonomy Groups and Calibrated Geometry (Oxford University Press, 2007).CrossRefGoogle Scholar
[12]Kovalev, A.. Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565 (2003), 125–160.Google Scholar
[13]Kovalev, A. and Nordström, J.. Asymptotically cylindrical 7-manifolds of holonomy G 2 with applications to compact irreducible G 2-manifolds. Ann. Global Anal. Geom. 38 (2010), 221–257.CrossRefGoogle Scholar
[14]Lenstra, A. K., Lenstra, H.W. and Lovasz, L.. Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.CrossRefGoogle Scholar
[15]Mori, S. and Mukai, S.. Classification of Fano 3-folds with B 2 ≥ 2. Manuscripta Math. 36 (1981/1982), 147–162.CrossRefGoogle Scholar
[16]Mori, S. and Mukai, S.. Erratum: Classification of Fano 3-folds with B 2 ≥ 2. Manuscripta Math. 110 (2003), 407.CrossRefGoogle Scholar
[17]Nikulin, V. V.. Finite groups of automorphisms of Kählerian K3 surfaces. Trans. Moscow Math. Soc. 2 (1980), 71–135.Google Scholar
[18]Nikulin, V. V.. Integer symmetric bilinear forms and some of their applications. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238. English translation: Math. USSR Izvestia 14 (1980), 103–167.Google Scholar
[19]Nikulin, V. V.. Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. Algebro-geometric applications. J. Soviet Math. 22 (1983), 1401–1476.CrossRefGoogle Scholar
[20]Salamon, S.. Riemannian geometry and holonomy groups. Pitman Res. Notes in Math. 201 (Longman, Harlow, 1989).Google Scholar
[21]Voisin, C.. Miroirs et involutions sur les surfaces K3. Journées de Géometrie Algébrique d'Orsay (Orsay, 1992). Astérisque 218 (1993), 273–323.Google Scholar