Annales de l’institut Fourier (AIF)

Filip, Simion

Tosatti, Valentino

Smooth and Rough Positive Currents

Lenzhen, Anna

Souto, Juan

Variations on a theorem of Birman and Series

Bettiol, Renato G.

Piccione, Paolo

Infinitely many solutions to the Yamabe problem on noncompact manifolds

Lê, Thang T. Q.

Growth of homology torsion in finite coverings and hyperbolic volume

Pol, Delphine

On the values of logarithmic residues along curves

Li, Long

Subharmonicity of conic Mabuchi’s functional, I

Hashizume, Kenta

Minimal model theory for relatively trivial log canonical pairs

Esnault, Hélène

Shiho, Atsushi

Convergent isocrystals on simply connected varieties

Ohsawa, Takeo

On the local pseudoconvexity of certain analytic families of $\protect \mathbb{C}$

Codogni, Giulio

Dervan, Ruadhaí

AN
N
A
L
E
S
D
E
L’INSTI
T
U
T
F
O
U
R
IE
R
ANNALES
DE
L’INSTITUT FOURIER
Giulio CODOGNI & Ruadhaí DERVAN
Corrigendum to “Non-reductive automorphism groups, the Loewy filtration
and K-stability”
Tome 68, no 3 (2018), p. 1121-1123.
<http://aif.cedram.org/item?id=AIF_2018__68_3_1121_0>
© Association des Annales de l’institut Fourier, 2018,
Certains droits réservés.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE.
http://creativecommons.org/licenses/by-nd/3.0/fr/
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions générales
d’utilisation (http://aif.cedram.org/legal/).
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
68, 3 (2018) 1121-1123
CORRIGENDUM TO “NON-REDUCTIVE
AUTOMORPHISM GROUPS, THE LOEWY
FILTRATION AND K-STABILITY”
Annales de l’institut Fourier, vol. 66 (2016), n°5, 1895–1921
by Giulio CODOGNI & Ruadhaí DERVAN
There is an error in the proof of [2, Lemma 2.4], where the multiplica-
tivity of the Loewy filtration is claimed (pointed out to us by Y. Odaka).
We are not aware of any examples in which the Loewy filtration of a po-
larised variety is not multiplicative, and indeed in all examples given it is
multiplicative. Moreover, multiplicativity is always true in the toric case;
we briefly sketch the argument below.
The error is as follows. To start with, the results of [1] used do not apply
in our situation. This in itself is not a serious issue, as the Jacobson ideal
can be replaced by the ideal generated by the image of the Lie algebra of
unipotent radical U of Aut(X, L) inside the universal enveloping algebra of
gl(H0(X, kL)). The main problem is that the containment
J(A)iR` · J(A)jRs ⊂ J(A)i+jR`+s
does not hold in the general. Indeed, it is not true that
(0.1) (g∗s)(h∗t) = (gh)∗st
where g and h are in U , and s and t are sections.
The proof of multiplicativity in the toric case is as follows (which arose
from discussions with H. Süss). Let T ⊂ Aut(X, L) be a maximal torus. As
the Loewy filtration is equivariant with respect to the action of Aut(X, L),
it is enough to prove the claim when s and t are eigenvectors of T , g and
h are in U , and g = exp(A) and h = exp(B), with A and B eigenvectors of
Keywords: K-stability, reductive groups, Kähler–Einstein metrics, radical filtration.
1122 Giulio CODOGNI & Ruadhaí DERVAN
T . In this case, both sides of equation (0.1) are eigenvectors for the action
of T with the same eigenvalue. A key feature of toric varieties is that the
eigenspaces are one dimensional, so equation (0.1) is true up to a scalar,
and this is enough to prove the multiplicativity of the Loewy filtration. We
plan to return to the Loewy filtration in the toric setting in future work.
While we are not aware of any examples in which the Loewy filtra-
tion of a polarised variety is not multiplicative, there is a related example
provided to us by Y. Odaka in which multiplicativity fails. Consider P3
with homogeneous coordinates x0, x1, x2, x3 with an action of a ∈ C+ by
x0 → x0, x1 → x1, x2 → x2+ax0, x3 → x3+ax1. It is then a straightforward
calculation to show that the induced Loewy filtration is not multiplicative.
Remark that the Loewy filtration induced from Aut(P3,O(1)) is trivial
since its automorphism group is reductive.
In general the Loewy filtration still produces a sequence of test configu-
rations; it may be that Conjecture B should be modified to state that the
Loewy filtration destabilises in a suitable sense of uniform K-polystability.
There is also another smaller error in [2, Section 1.2]. First, we did not
remark that the polynomial Donaldson–Futaki invariant introduced in Def-
inition 1.4 actually corresponds to the Chow∞-weight used for example
in [3]. As explained in [3], this invariant is equal to the Donaldson–Futaki
invariant when the filtration is finitely generated, but it is not clear if there
is any inequality in the general case. An example in which the two invariants
differ is [3, Example 4].
In particular, the proof of [2, Lemma 1.14] is incorrect. The mistake in
the proof is that the weight function is equal to the weight polynomial
w(k) just for sufficiently large k, and this threshold can actually grow with
the r appearing in the claim of the Lemma. This also invalidates [2, Theo-
rem 1.10], and item (ii) of [2, Corollary 1.14] is most likely not equivalent
to (i) and (iii).
This does not affect any of the examples, as the filtration is finitely
generated in each case. Finite generation follows as the filtration is by
vanishing order in each example (along the fixed locus of the unipotent
radical of the automorphism group), and the argument of Proposition 3.1
implies vanishing order filtrations are finitely generated on toric varieties.
BIBLIOGRAPHY
[1] I. Assem, D. Simson & A. Skowroński, Elements of the representation theory of as-
sociative algebras. Vol. 1: Techniques of representation theory, London Mathematical
Society Student Texts, vol. 65, Cambridge University Press, 2006, x+458 pages.
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM 1123
[2] G. Codogni & R. Dervan, “Non-reductive automorphism groups, the Loewy filtra-
tion and K-stability”, Ann. Inst. Fourier 66 (2016), no. 5, p. 1895-1921.
[3] G. Székelyhidi, “Filtrations and test-configurations”, Math. Ann. 362 (2015), no. 1-
2, p. 451-484, With an appendix by Sebastien Boucksom.
Manuscrit reçu le 25 janvier 2018,
révisé le 5 février 2018,
accepté le 5 février 2018.
Giulio CODOGNI
EPFL SB MATH,
Station 8,
CH-1015 Lausanne (Switzerland)
giulio.codogni@epfl.ch
Ruadhaí DERVAN
École Polytechnique,
Cour Vaneau
91120 Palaiseau (France)
r.dervan@dpmms.cam.ac.uk
TOME 68 (2018), FASCICULE 3


Corrigendum to “Non-reductive automorphism groups, the Loewy filtration and K-stability”

Not a member yet