Optimal symplectic connections on holomorphic submersions

Dervan, R. and Sektnan, L. M. (2021) Optimal symplectic connections on holomorphic submersions. Communications on Pure and Applied Mathematics, 74(10), pp. 2132-2184. (doi: 10.1002/cpa.21930)

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The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation. We begin with a smooth fibration for which all fibres admit a constant scalar curvature Kähler metric. When the fibres admit automorphisms, such metrics are not unique in general, but rather are unique up to the action of the automorphism group of each fibre. We define an equation which, at least conjecturally, determines a canonical choice of constant scalar curvature Kähler metric on each fibre. When the fibration is a projective bundle, this equation specialises to asking that the hermitian metric determining the fibrewise Fubini-Study metric is Hermite-Einstein. Assuming the existence of an optimal symplectic connection and the existence of an appropriate twisted extremal metric on the base of the fibration, we show that the total space of the fibration itself admits an extremal metric for certain polarisations making the fibres small.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Dervan, Dr Ruadhaí
Authors: Dervan, R., and Sektnan, L. M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Communications on Pure and Applied Mathematics
ISSN (Online):1097-0312
Published Online:04 August 2020
Copyright Holders:Copyright © 2020 The Authors
First Published:First published in Communications on Pure and Applied Mathematics 74(10):2132-2184
Publisher Policy:Reproduced under a Creative Commons License

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