# Optimal symplectic connections on holomorphic submersions

## TL;DR

This paper introduces the optimal symplectic connection equation, a new geometric PDE, to construct extremal Kähler metrics on holomorphic submersions, unifying and extending previous results in the field.

## Contribution

It proposes a novel PDE called the optimal symplectic connection equation, providing a new method to obtain extremal Kähler metrics on total spaces of fibrations.

## Key findings

- Constructs extremal Kähler metrics on total spaces of certain fibrations.
- Unifies previous results by Hong, Fine, and others.
- Shows existence of extremal metrics under specific conditions.

## Abstract

The main result of this paper gives a new construction of extremal K\"ahler 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 K\"ahler 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 K\"ahler 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.