Hostname: page-component-669899f699-g7b4s Total loading time: 0 Render date: 2025-04-26T08:22:21.794Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness of optimal symplectic connections

Part of: Complex manifolds Global differential geometry

Published online by Cambridge University Press:  04 March 2021

Ruadhaí Dervan  and
Lars Martin Sektnan
Ruadhaí Dervan
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, United Kingdom; E-mail: R.Dervan@dpmms.cam.ac.uk
Lars Martin Sektnan
Affiliation:
Institut for Matematik, Aarhus University, 8000, Aarhus C, Denmark; E-mail: lms@math.au.dk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation.

We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

MSC classification

Primary: 53C55: Hermitian and Kählerian manifolds
Secondary: 32Q26: Notions of stability
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e18
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Anchouche, B. and Biswas, I., Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. Amer. J. Math. 123 (2001), no. 2, 207–228.CrossRefGoogle Scholar
Berman, R. and Berndtsson, B., Convexity of the $\;K$-energy on the space of Kähler metrics and uniqueness of extremal metrics. J. Amer. Math. Soc. 30 (2017), no. 4, 1165–1196.CrossRefGoogle Scholar
Darvas, T., The Mabuchi geometry of finite energy classes Adv. Math. 285 (2015), 182–219.CrossRefGoogle Scholar
Darvas, T. and Rubinstein, Y. A., Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics J. Amer. Math. Soc. 30 (2017), no. 2, 347–387.CrossRefGoogle Scholar
Datar, V. and Székelyhidi, G., Kähler-Einstein metrics along the smooth continuity method Geom. Funct. Anal. 26 (2016), no. 4, 975–1010.CrossRefGoogle Scholar
Dervan, R., Uniform stability of twisted constant scalar curvature Kähler metrics Int. Math. Res. Not. IMRN 2016, no. 15, 4728–4783.CrossRefGoogle Scholar
Dervan, R. and Naumann, P., Moduli of polarised manifolds via canonical Kähler metrics (2018), arXiv:1810.02576Google Scholar
Dervan, R. and Ross, J., K-stability for Kähler manifolds Math. Res. Lett. 24 (2017), no. 3, 689–739.CrossRefGoogle Scholar
Dervan, R. and Ross, J., Stable maps in higher dimensions Math. Ann. 374 (2019), no. 3-4, 1033–1073.CrossRefGoogle Scholar
Dervan, R. and Sektnan, L. M., Extremal metrics of fibrations Proc. Lond. Math. Soc. (3) 120 (2020), no. 4, 587–616.CrossRefGoogle Scholar
Dervan, R. and Sektnan, L. M., Optimal symplectic connections on holomorphic submersions Comm. Pure. Appl. Math. (to appear) (2019), arXiv:1907.11014Google Scholar
Dervan, R. and Sektnan, L. M., Moduli theory, stability of fibrations and optimal symplectic connections 2019, arXiv:1911.12701Google Scholar
Donaldson, S. K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26.CrossRefGoogle Scholar
Donaldson, S. K., Symmetric spaces, Kähler geometry and Hamiltonian dynamics. Northern California Symplectic Geometry Seminar, 13–33, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, (1999).CrossRefGoogle Scholar
Donaldson, S. K., Scalar curvature and projective embeddings. I J. Differential Geom. 59 (2001), no. 3, 479–522.CrossRefGoogle Scholar
Fine, , Constant scalar curvature Kähler metrics on fibred complex surfaces J. Differential Geom. 68 (2004), no. 3, 397–432.CrossRefGoogle Scholar
Fine, J., Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle Math. Res. Lett. 14 (2007), no. 2, 239–247.CrossRefGoogle Scholar
Fujiki, A. and Schumacher, G., The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics Publ. Res. Inst. Math. Sci. 26 (1990), no. 1, 101–183.CrossRefGoogle Scholar
Futaki, A. and Mabuchi, T., Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Ann. 301 (1995), no. 2, 199–210.CrossRefGoogle Scholar
Gauduchon, P., Calabi’s Extremal Kähler Metrics: An Elementary Introduction. Book available online.Google Scholar
Hashimoto, Y., Existence of twisted constant scalar curvature Kähler metrics with a large twist Math. Z. 292 (2019), no. 3-4, 791–803.CrossRefGoogle Scholar
Hong, Y, J., Constant Hermitian scalar curvature equations on ruled manifolds J. Differential Geom. 53 (1999), no. 3, 465–516.CrossRefGoogle Scholar
Inoue, E., The moduli space of Fano manifolds with Kähler-Ricci solitons Adv. Math. no. 357 (2019). 1–65.CrossRefGoogle Scholar
Keller, J., Twisted balanced metrics Lie Groups: New research, Mathematics Research Developments (2009), 267–281.Google Scholar
Mabuchi, T., K-energy maps integrating Futaki invariants Tohoku Math. J. (2) 38 (1986), no. 4, 575–593.CrossRefGoogle Scholar
Mabuchi, T., Some symplectic geometry on compact Kähler manifolds. I Osaka J. Math. 24 (1987), no. 2, 227–252.Google Scholar
McDuff, D. and Salamon, D., Introduction to symplectic topology, 3rd ed., Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford (2017). xi+623 pp.CrossRefGoogle Scholar
Michor, P., Topics in differential geometry Graduate Studies in Mathematics, vol. 93, American Mathematical Society, Providence, RI (2008). xii+494 pp.Google Scholar
Székelyhidi, G., An introduction to extremal Kähler metrics Graduate Studies in Mathematics, 152. American Mathematical Society, Providence, RI (2014). xvi+192 pp.CrossRefGoogle Scholar
Tian, G., Canonical metrics in Kähler geometry Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2000). Notes taken by Meike Akveld. vi+101 pp.CrossRefGoogle Scholar