Uniqueness of optimal symplectic connections

Abstract

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.

Extremal metrics on the total space of destabilising test configurations

“…The remaining argument follows exactly similar arguments in e.g. [21,23]. The statement below is just rephrasing Theorem 1, using the parameter ε instead of k.…”

“…In fact, as used e.g. in [21,Remark 3.8], the actual solutions can be ensured to be an O(ε κ−2δ ) perturbation of ε,κ . In particular, we can, for any κ, choose a κ > κ such that the actual solutions produced from ε,κ agree with ε,κ to order ε κ .…”

Stability conditions for polarised varieties

“…It only remains to show that x ∈ (C * ) 2 . For this, we denote z = (x, y) and note that there is a constant C > 0 such that x − λ 1 ε p1/2 ≤ Cε (p1+1)/2 ; this is a standard consequence of the quantitative inverse function theorem, see for example [18,Remark 3.8]. In particular, taking ε small but positive shows that x ∈ R >0 .…”

Optimal Symplectic Connections and Deformations of Holomorphic Submersions

“…This seems to be the first geometric PDE in complex geometry which involves both curvature quantities and the change in complex structure. We expect optimal symplectic connections to be unique, as happens in the relatively cscK case [9,20], up to the action of a suitable subset of the automorphisms of Y which preserves the projection π Y . In this way, one can genuinely call an optimal symplectic connection a canonical choice of a relatively Kähler metric on a relatively K-semistable fibration.…”