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Optimal Symplectic Connections on Holomorphic Submersions

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Dervan, Ruadhaí 
Sektnan, Lars Martin 


jats:pThe 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.</jats:p>jats:pWe 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.</jats:p>jats:pAssuming 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. © 2021 The Authors. jats:italicCommunications on Pure and Applied Mathematics</jats:italic> published by Wiley Periodicals LLC.</jats:p>



4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences

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Communications on Pure and Applied Mathematics

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London Mathematical Society, International Centre for Mathematical Sciences, CIRGET