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Non-reductive automorphism groups, the Loewy filtration and K-stability. (Groupes d’automorphismes non réductifs, filtration de Loewy et K-stabilité.) (English. French summary) Zbl 1370.32010
Ann. Inst. Fourier 66, No. 5, 1895-1921 (2016); corrigendum ibid. 68, No. 3, 1121-1123 (2018).
Let \((X,L)\) be a polarised variety and let \(R(X,L)=\bigoplus_{k\geq 0} H^0(X,L^k)\) be the coordinate ring of \((X,L)\). An algebro-geometric concept of K-polystability is defined associating to each filtration of \(R(X,L)\) a weight: the Donaldson-Futaki invariant. A canonical filtration of \(R(X,L)\) is constructed in the case \(X\) has non-reductive automorphism group: Loewy filtration. It is proved that the Loewy filtration is admissible, is \(\mathrm{Aut}(X,L)\)-equivariant and is trivial if and only if \(\mathrm{Aut}(X,L)\) is reductive. In particular K-polystability can be studied by this fibration. It is conjectured that in the case \(\mathrm{Aut}(X,L)\) is non-reductive the Loewy filtration destabilises \((X,L)\). The conjecture is proved in several interesting cases, in particular this provides examples of polarised varieties without constant scalar curvature Kähler metrics. The conjecture in this paper is related to the proof of the Yau-Tian-Donaldson conjecture. Also this paper gives a new method of destabilising varieties.

MSC:
32Q26 Notions of stability for complex manifolds
32M99 Complex spaces with a group of automorphisms
32Q20 Kähler-Einstein manifolds
17B20 Simple, semisimple, reductive (super)algebras
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