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It is known that the automorphism group of a K-polystable Fano manifold isreductive. Codogni and Dervan construct a canonical filtration of the sectionring, called Loewy filtration, and conjecture that the Loewy filtrationdestabilizes any Fano variety with non-reductive automorphism group. In thisnote, we give a counterexample to their conjecture.

Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups