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Applications of the moduli continuity method to log K-stable pairs

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  • Patricio Gallardo, University of California at Riverside, Washington University St. Louis
    • ,
  • Jesus Martinez-Garcia, University of Essex
    • ,
  • Cristiano Spotti

The ‘moduli continuity method’ permits an explicit algebraisation of the Gromov–Hausdorff compactification of Kähler–Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the ‘log setting’ to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from geometric invariant theory (GIT). More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.

OriginalsprogEngelsk
TidsskriftJournal of the London Mathematical Society
Vol/bind103
Nummer2
Sider (fra-til)729-759
Antal sider31
ISSN0024-6107
DOI
StatusUdgivet - mar. 2021

Bibliografisk note

Funding Information:
PG is grateful for the working environment of the Department of Mathematics in Washington University at St. Louis. PG's travel related to this project was partially covered by the FRG Grant DMS-1361147 (P.I Matt Kerr). This project was started at the Hausdorff Research Institute for Mathematics (HIM) during a visit by the authors as part of the Research in Groups project Moduli spaces of log del Pezzo pairs and K-stability. We thank HIM for their generous?support. We would like to thank F. Gounelas for a clarification regarding the Fano index in deformation families. We would like to thank M. de Borb?n, J. Nordstr?m, Y. Odaka and S. Sun for useful?comments.

Publisher Copyright:
© 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

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