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Non-reductive automorphism groups, the Loewy filtration and K-stability

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Codogni, Giulio 
Dervan, Ruadhaí 


We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue of Matsushima’s theorem regarding the existence of constant scalar curvature Kähler metrics. As an application, we give an example of an orbifold del Pezzo surface without a Kähler-Einstein metric.



K-stability, reductive groups, Kähler-Einstein metrics, radical filtration

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Annales de l'Institut Fourier

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l'Institut Fourier

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This project started during the school “Minicourses on Stability” at the University of Coimbra in April 2014; we thank the organisers for the stimulating environment. We would like to thank Giovanni Cerulli Irelli, Jesus Martinez Garcia, Julius Ross, Roberto Svaldi, Filippo Viviani and Xiaowei Wang for useful discussions. Both authors would especially like to thank David Witt Nystr¨om and Jacopo Stoppa for several discussions on the present work. We thank the referee for several helpful comments. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 307119. GC was funded by the grants FIRB 2012 “Moduli Spaces and Their Applications” and by the ERC StG 307119 - StabAGDG. RD was funded by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1) and a Fondation Wiener-Anspach Scholarship.