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Moduli theory, stability
of fibrations and optimal symplectic connections
Ruadhaí Dervan and Lars Martin Sektnan |
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Geometry & Topology 25 (2021) 2643–2697
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Abstract |
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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. |
Keywords
stability, Kähler geometry, fibrations, moduli
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Mathematical Subject Classification
Primary: 53C55
Secondary: 14D06, 14D20
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References |
Publication
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
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Authors |
| Ruadhaí Dervan | |
| DPMMS, Centre for Mathematical
Sciences University of Cambridge Cambridge United Kingdom |
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| Lars Martin Sektnan | |
| Institut for Matematik Aarhus University Aarhus Denmark |
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