Optimal Symplectic Connections and Deformations of Holomorphic Submersions

Abstract

We give a general construction of extremal Kähler metrics on the total space of certain holomorphic submersions, extending results of Dervan-Sektnan, Fine, and Hong. We consider submersions whose fibres admit a degeneration to Kähler manifolds with constant scalar curvature, in a way that is compatible with the fibration structure. Thus we allow fibres that are K-semistable, rather than K-polystable; this is crucial to moduli theory. On these fibrations we phrase a partial differential equation whose solutions, called optimal symplectic connections, represent a canonical choice of a relatively Kähler metric. We expect this to be the most general construction of a canonical relatively Kähler metric provided all input is smooth. We use the notion of an optimal symplectic connection and the geometry related to it to construct Kähler metrics with constant scalar curvature and extremal metrics on the total space, in adiabatic classes.

Canonical metrics on holomorphic fibre bundles

“…This setting is now genuinely outside the realm of Hermite-Einstein metrics, and our main result demonstrates that, at least in the case of smooth fibrations, the optimal symplectic connection equation should be thought of as a generalisation of the Hermite-Einstein equation to the setting of fibrations where the complex structure of the fibre varies. In this nonisotrivial setting it is important to modify the equation in order to allow strictly K-semistable fibres, as was observed by Ortu [Ort22]. It is expected that our main result should find applications to the study of optimal symplectic connections on non-isotrivial deformations of isotrivial fibrations, as can appear in the work of Ortu.…”