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Uniqueness of optimal symplectic connections

Part of: Global differential geometry Complex manifolds

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

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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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