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

Patricio Gallardo, Jesus Martinez-Garcia, Cristiano Spotti*

*Corresponding author af dette arbejde

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Abstract

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

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