×

Twisted connected sums and special Riemannian holonomy. (English) Zbl 1043.53041

The author presents a new method for constructing Riemannian metrics with holonomy \(G_2\) on compact \(7\)-dimensional manifolds. Based on some work by Tian and Yau the author first constructs a class of noncompact asymptotically cylindrical Riemannian manifolds with holonomy \(SU(3)\). The Riemannian product of such a manifold and a circle \(S^1\) has again holonomy \(SU(3)\) and an end that is asymptotic to a half-cylinder. A cross-section of this half-cylinder is the Riemannian product of two circles and a complex surface of type K3 with a hyper-Kähler metric. By truncating the ends of two such asymptotically cylindrical \(7\)-manifolds, cutting off to the cylindrical metric and identifying the boundaries by an orientation-reversing isometry that interchanges the two \(S^1\) factors he obtains a closed compact Riemannian \(7\)-manifold. The twisting that is built into the isometry identifying the boundaries has the purpose to avoid an infinite fundamental group.
The construction is based on a gluing theorem for appropriate elliptic partial differential equations and yields compact \(7\)-manifolds with a \(G_2\)-structure. The author then discusses the topology of the resulting \(7\)-manifolds when one starts the construction with Fano manifolds, which leads to many new topological types of compact \(G_2\)-manifolds.

MSC:

53C29 Issues of holonomy in differential geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
PDF BibTeX XML Cite
Full Text: DOI arXiv Link

References:

[1] Bryant, On the construction of some complete metrics with exceptional holonomy, Math J 58 pp 829– (1989) · Zbl 0681.53021
[2] Kobayashi, On compact Ka hler manifolds with positive definite Ricci tensor, Math 74 pp 570– (1961)
[3] Shokurov, Smoothness of a general anticanonical divisor on a Fano variety Nauk SSSR transl, Mat English Math USSR Izv 43 pp 430– (1979)
[4] Atiyah, and Spectral asymmetry and Riemannian geometry, Math Proc Phil Soc 77 pp 97– (1975)
[5] Brandhuber, and Gauge theory at large N and new holonomy met - rics, Phys B pp 611– (2001)
[6] Bryant, Metrics with exceptional holonomy, Math 126 pp 525– (1987) · Zbl 0637.53042
[7] Joyce, Compact Riemannian - manifolds with holonomy II, Geom 7 pp 291– (1996) · Zbl 0861.53023
[8] Lockhart, Elliptic di erential operators on noncompact manifolds Norm Sup Pisa, Sci 12 pp 409– (1985)
[9] Salamon, Riemannian geometry and holonomy groups Longman, Pitman Res Notes Math pp 201– (1989) · Zbl 0685.53001
[10] Berger, and Le spectre d une varie te riemannienne Springer - Verlag, Notes Math pp 194– (1971)
[11] Kodaira, Complex manifolds and deformation of complex structures Springer - Verlag Special Riemannian holonomy Gluing theorems for complete anti - self - dual spaces, Funct Anal 11 pp 1229– (1986)
[12] Nikulin, Integer symmetric bilinear forms and some of their geometric applications SSSR transl, Mat English Math USSR Izv 43 pp 111– (1979)
[13] Iskovskih, Fano threefolds II Nauk SSSR and transl and, Mat English Math USSR Izv 41 pp 516– (1977)
[14] Tian, Complete Ka hler manifolds with zero Ricci curvature II, Math 106 pp 27– (1991)
[15] Tian, Complete Ka hler manifolds with zero Ricci curvature, Amer Math Soc 3 pp 579– (1990)
[16] Bott, On a theorem of Lefschetz Michigan, Math 6 pp 211– (1959) · Zbl 0113.36502
[17] Ciliberto, and Projective degenerations of surfaces Gaussian maps and Fano threefolds, Math 114 pp 641– (1993) · Zbl 0807.14028
[18] Floer, Self - dual conformal structures on lCP, Geom 2 pp 551– (1991) · Zbl 0736.53046
[19] Harvey, Calibrated geometries, Acta Math 148 pp 47– (1982) · Zbl 0584.53021
[20] Iskovskikh, Fano varieties Algebraic geometry V Springer - Verlag, Math Sci pp 47– (1999) · Zbl 0912.14013
[21] Ciliberto, and Classification of varieties with canonical curve section via Gaus - sian maps on canonical curves, Math 120 pp 1– (1998) · Zbl 0934.14028
[22] Comm, r Real Killing spinors and holonomy, Math Phys pp 154– (1993)
[23] Mukai, Classification of - folds with, Math 36 pp 147– (1981) · Zbl 0478.14033
[24] Welters, Polarized abelian varieties and the heat equations, Math 49 pp 173– (1983) · Zbl 0576.14042
[25] Chong, and General metrics of holo - nomy and contraction limits, Phys B pp 638– (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.