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New \(\mathrm{G}_2\)-holonomy cones and exotic nearly Kähler structures on \(S^6\) and \(S^3\times S^3\). (English) Zbl 1381.53086
This astonishing paper solves a longstanding problem in the field of ‘special geometry’, by exhibiting the first examples of simply connected, non-homogeneous nearly Kähler structures on the manifolds \(S^6\) and \(S^3\times S^3\), invariant by actions of \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) of cohomogeneity one.
Nearly Kähler (NK) manifolds \(N\) probably arose from the work of A. Gray on holonomy, see [Math. Ann. 223, 233–248 (1976; Zbl 0345.53019)], and J. A. Wolf and A. Gray’s \(3\)-symmetric spaces [J. Differ. Geom. 2, 115–159 (1968; Zbl 0182.24702)]. They form the class of almost Hermitian manifolds that are closest to being Kähler in the language of the Gray-Salamon intrinsic torsion of \(G\)-structures, see [S. Salamon, Riemannian geometry and holonomy groups. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons (1989; Zbl 0685.53001); Milan J. Math. 71, 59–94 (2003; Zbl 1055.53039)]. They are Einstein with positive scalar curvature, and compact with finite fundamental group if complete. A deRham-like structure theorem tells us \(6\)-dimensional ones are distinguished [P.-A. Nagy, Asian J. Math. 6, No. 3, 481–504 (2002; Zbl 1041.53021)].
As a matter of fact, it is the connection to \(G_2\) geometry that is key: the metric cone over an NK \(6\)-manifold has a holonomy \(G_2\) [C. Bär, Commun. Math. Phys. 154, No. 3, 509–521 (1993; Zbl 0778.53037)]. This is of interest also because it ties in nicely with the general theory of Killing spinors [H. Baum et al., Twistors and Killing spinors on Riemannian manifolds. Stuttgart etc.: B.G. Teubner Verlagsgesellschaft (1991; Zbl 0734.53003)], see also [I. Agricola et al., J. Geom. Phys. 98, 535–555 (2015; Zbl 1333.53037)].
The scarcity of NK \(6\)-manifolds (i.e., \(G_2\)-holonomy cones) is surprising and somehow vexing, especially if compared to other special geometries: there are in fact infinitely many Calabi-Yau, hyperKähler and \(\mathrm{Spin}(7)\)-cones. Besides, it is known that \(S^6, S^3\times S^3, \mathbb{CP}^3\) and the full flag \(F^3=\operatorname{SU}(3)/(S^1\times S^1)\) are the only homogeneous NK \(6\)-manifolds [J.-B. Butruille, Ann. Global Anal. Geom. 27, No. 3, 201–225 (2005; Zbl 1079.53044)], which opens the question of complete, non-homogeneous instances.
There are two natural ways to find NK \(6\)-manifolds, namely imposing enough symmetry (cohomogeneity-one) in order to attain manageable differential equations, and resolving singular NK examples. The authors combined the two techniques: First they take the sine cone of a Sasaki-Einstein \(5\)-manifold to obtain a singular NK space with two isolated singularities modelled on a Calabi-Yau cone. Then they desingularise it as a cohomogeneity-one space by replacing said singularities with conical Calabi-Yaus.
The authors take the potential models of complete cohomogeneity-one NK \(6\)-manifolds given by F. Podestà and A. Spiro [J. Geom. Phys. 60, No. 2, 156–164 (2010; Zbl 1184.53074); Commun. Math. Phys. 312, No. 2, 477–500 (2012; Zbl 1262.53062)], and ‘match’, instead of glueing, the local geometries. They study carefully the evolution ODEs of one-parameter family of nearly hypo structures [D. Conti and S. Salamon, Trans. Am. Math. Soc. 359, No. 11, 5319–5343 (2007; Zbl 1130.53033); M. Fernández et al., J. Lond. Math. Soc., II. Ser. 78, No. 3, 580–604 (2008; Zbl 1158.53018)] and find a way to recognise which local solutions extend to complete metrics.
The conjecture posed in the paper posits that are there no more (inhomogeneous) cohomogeneity-one NK structures on simply connected six-manifolds.

MSC:
53C29 Issues of holonomy in differential geometry
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