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Moduli theory, stability of fibrations and optimal symplectic connections

Ruadhaí Dervan and Lars Martin Sektnan

Geometry & Topology 25 (2021) 2643–2697

K–polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler 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 Kähler 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 Kähler 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.

stability, Kähler geometry, fibrations, moduli
Mathematical Subject Classification
Primary: 53C55
Secondary: 14D06, 14D20
Received: 6 April 2020
Revised: 16 June 2020
Accepted: 23 July 2020
Published: 3 September 2021
Proposed: Richard P Thomas
Seconded: Frances Kirwan, Simon Donaldson
Ruadhaí Dervan
DPMMS, Centre for Mathematical Sciences
University of Cambridge
United Kingdom
Lars Martin Sektnan
Institut for Matematik
Aarhus University