Optimal Symplectic Connections on Holomorphic Submersions

Abstract

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. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Uniqueness of optimal symplectic connections

“…Note that when the fibres admit continuous automorphisms, there is an infinite-dimensional family of relatively Kähler metrics, which are cscK on each fibre. We conjectured that solutions of the optimal symplectic connection equation are unique, meaning that optimal symplectic connections do give a canonical choice of ∈ when they exist [11,Conjecture 1.2]. Here we prove that conjecture.…”

“…For Proposition 3.1 to construct 'approximate twisted extremal metrics', by Lemma 2.1 one needs to be a holomorphy potential with respect to . The following simple lemma establishes this and is implicit in [11]. The lemma is proven explicitly in the case and are projective in [12,Proposition 3.11]; the proof given there also applies to more singular algebro-geometric fibrations.…”

“…Thus can be thought of as parametrising holomorphic vector fields on ( , ). Going further, one can naturally define a smooth vector bundle → associated with the submersion : → with fibre [11,Section 3.1]; this is the step at which our assumption that dim is independent of ∈ enters. Thus a section of over corresponds to a function on whose restriction to each fibre is a mean-value zero holomorphy potential with respect to .…”

“…This also defines a natural splitting of the space ∞ ( , R) [11,Section 3.1] into three components; one is ∞ ( , R) described above. induces a projection…”

“…In previous work, we introduced a candidate answer for our motivating question, in the form of an optimal symplectic connection [11]. This is a relatively Kähler metric ∈ , which restricts to a cscK metric on each fibre and satisfies an additional geometric partial differential equation described explicitly in Section 2.2.…”