Extremal metrics on fibrations

Dervan, Ruadhaí; Sektnan, Lars Martin

doi:10.1112/plms.12297

Cited by 11 publications

(43 citation statements)

References 54 publications

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“…Proof. This is proved identically to [10,Proposition 5.6] and [11,Proposition 4.11], with parts (i) and (ii) building heavily on the lower-dimensional work of Fine [16,Section 3.3]). The only difference with [11,Proposition 4.11] is the behaviour on functions ∈ ∞ ( , R).…”

Section: The Approximate Solutionmentioning

confidence: 71%

“…→ . Uniqueness statements for twisted extremal metrics are proved in [24,2,10], and when Aut 0 ( ) is the identity, combined with Theorem 4 and our new adiabatic limit techniques, are enough to prove uniqueness of optimal symplectic connections. In the case Aut 0 ( ) is non-trivial, we have to work harder and employ techniques concerning the action of the automorphism group on the space of Kähler potentials developed in the important work of Darvas-Rubinstein [4].…”

Section: Theorem 4 Suppose Is An Optimal Symplectic Connection Thenmentioning

confidence: 99%

“…The uniqueness result is originally due to Keller in the case that either Aut( , ) is discrete or is positive, with = 1 ( ) the first Chern class of an ample line bundle [24]. In general the uniqueness statement follows from the work of Berman-Berndtsson [2], and the geometric version of the uniqueness statement can be found as [10,Corollary 3.8]. We will require, and hence will prove, more precise uniqueness statements in Section 4.…”

Section: Ifmentioning

confidence: 99%

“…We will require, and hence will prove, more precise uniqueness statements in Section 4. One can also find results concerning reductivity of the relevant Lie algebras of holomorphic vector fields in [10,Theorem 4.1] and [5, Proposition 7]; these will not be needed in the present work.…”

Section: Ifmentioning

confidence: 99%

“…Twisted extremal metrics are best viewed as canonical metrics on maps between complex manifolds [10,9]; in our case, the map is :…”

Section: Theorem 4 Suppose Is An Optimal Symplectic Connection Thenmentioning

confidence: 99%

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Uniqueness of optimal symplectic connections

Dervan¹,

Sektnan

2021

Forum of Mathematics, Sigma

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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.

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Section: The Approximate Solutionmentioning

confidence: 71%

Section: Theorem 4 Suppose Is An Optimal Symplectic Connection Thenmentioning

confidence: 99%

Section: Ifmentioning

confidence: 99%

Section: Ifmentioning

confidence: 99%

“…Twisted extremal metrics are best viewed as canonical metrics on maps between complex manifolds [10,9]; in our case, the map is :…”

Section: Theorem 4 Suppose Is An Optimal Symplectic Connection Thenmentioning

confidence: 99%

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Uniqueness of optimal symplectic connections

Dervan¹,

Sektnan

2021

Forum of Mathematics, Sigma

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Optimal Symplectic Connections on Holomorphic Submersions

Dervan¹,

Sektnan

2020

Comm Pure Appl Math

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

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