Moduli theory, stability of fibrations and optimal symplectic connections

Dervan, R. and Sektnan, L. M. (2021) Moduli theory, stability of fibrations and optimal symplectic connections. Geometry and Topology, 25(5), pp. 2643-2697. (doi: 10.2140/gt.2021.25.2643)

[img] Text
306100.pdf - Accepted Version



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.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Dervan, Dr Ruadhaí
Authors: Dervan, R., and Sektnan, L. M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Geometry and Topology
Publisher:Mathematical Sciences Publishers
ISSN (Online):1364-0380
Copyright Holders:Copyright © 2021 Mathematical Sciences Publishers
First Published:First published in Geometry and Topology 25(5):2643–2697
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

University Staff: Request a correction | Enlighten Editors: Update this record