Moduli theory, stability of fibrations and optimal symplectic connections

TL;DR

This paper introduces a stability notion for fibrations of K-polystable varieties, linking it to K"ahler geometry and optimal symplectic connections, and explores their equivalence through GIT and moment map analysis.

Contribution

It extends classical slope stability to families of K-polystable varieties and connects this to the existence of optimal symplectic connections.

Findings

Existence of optimal symplectic connection implies semistability of the fibration.

Finite dimensional GIT problem characterizes stability and existence of zero of a moment map.

Conjecture that optimal symplectic connection existence is equivalent to polystability.

Abstract

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical K\"ahler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations. Our main result relates this stability condition to K\"ahler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature K\"ahler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.