The CR geometry of weighted extremal Kähler and Sasaki metrics

Abstract

We establish an equivalence between conformally Einstein–Maxwell Kahler 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 Kahler 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 Kahler 4-manifolds in the sense of Calabi (Extremal Kahler 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 Kahler metrics arise as transversal Kahler 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 Kahler metric (Apostolov et al. in Levi–Kahler reduction of CR structures, product of spheres, and toric geometry, arXiv:1708.05253 ; Apostolov et al. in Weighted extremal Kahler 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 Kahler 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 Kahler 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 Kahler cone corresponding to a Sasaki polarization, studied in Boyer and van Coevering (Math Res Lett 25:1–19, 2018) and Collins and Szekelyhidi (J Differ Geom 109:81–109, 2018).