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The CR geometry of weighted extremal Kähler and Sasaki metrics
Abstract
We establish an equivalence between conformally Einstein–Maxwell Kähler 4-manifolds recently studied in Apostolov et al. (J Reine Angew Math 721:109–147, 2016), Apostolov and Maschler (J Eur Math Soc 21:1319–1360, 2019), Futaki and Ono (J Math Soc Jpn 70:1493–1521, 2018), Koca et al. (Ann Glob Anal Geom 50, 29–46, 2016), LeBrun (Einstein–Maxwell equations, extremal Kähler metrics, and Seiberg–Witten theory in “The Many Facets of Geometry: A Tribute to Nigel Hitchin”, Oxford University Press, Oxford, pp 17–33, 2009), LeBrun (J Geom Phys 91:–171, 2015) and LeBrun (Commun Math Phys 344:621–653, 2016) and extremal Kähler 4-manifolds in the sense of Calabi (Extremal Kähler metrics, seminar on differential geometry, annals of mathematics studies, vol 102, pp 259–290, Princeton University Press, Princeton, 1982) with nowhere vanishing scalar curvature. The corresponding pairs of Kähler metrics arise as transversal Kähler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki–Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kähler metric (Apostolov et al. in Levi–Kähler reduction of CR structures, product of spheres, and toric geometry, arXiv:1708.05253; Apostolov et al. in Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles, arXiv:1808.02813; Lahdili in J Geom Anal 29:542–568, 2019; Lahdili in Int Math Res Not, arXiv:1710.00235; Lahdili in Proc Lond Math Soc 119:1065–1114, 2019) illuminating and uniting several explicit constructions in Kähler and Sasaki geometry. It also leads to new existence and non-existence results for extremal Sasaki metrics, suggesting a link between the notions of relative weighted K-stability of a polarized variety introduced in Apostolov et al. (Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles, arXiv:1808.02813) and Lahdili (Proc Lond Math Soc 119:1065–1114, 2019), and relative K-stability of the Kähler cone corresponding to a Sasaki polarization, studied in Boyer and van Coevering (Math Res Lett 25:1–19, 2018) and Collins and Székelyhidi (J Differ Geom 109:81–109, 2018).