Let (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : X → X and η : X → Y, both isomorphic onX X̂ 0, such that [...] whereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor onX such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way that [...] where C1 and C2 are positive real constants independent of the choice of μ and ℓ. This plays an important role in our forthcoming papers on the existence of constant scalar curvature Kähler metrics (cf. [6]) and also on the compactified moduli space of test configurations (cf. [5],[7]).
@article{bwmeta1.element.doi-10_1515_coma-2016-0005,
author = {Toshiki Mabuchi},
title = {An l-th root of a test configuration of exponent l},
journal = {Complex Manifolds},
volume = {3},
year = {2016},
zbl = {1339.32012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2016-0005}
}
Toshiki Mabuchi. An ℓ-th root of a test configuration of exponent ℓ. Complex Manifolds, Tome 3 (2016) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2016-0005/