# An ℓ-th root of a test configuration of exponent ℓ

Complex Manifolds (2016)

- Volume: 3, Issue: 1, page 169-185
- ISSN: 2300-7443

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topToshiki Mabuchi. "An ℓ-th root of a test configuration of exponent ℓ." Complex Manifolds 3.1 (2016): 169-185. <http://eudml.org/doc/277104>.

@article{ToshikiMabuchi2016,

abstract = {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]).},

author = {Toshiki Mabuchi},

journal = {Complex Manifolds},

keywords = {polarized algebraic manifold; desingularizations; test configurations},

language = {eng},

number = {1},

pages = {169-185},

title = {An ℓ-th root of a test configuration of exponent ℓ},

url = {http://eudml.org/doc/277104},

volume = {3},

year = {2016},

}

TY - JOUR

AU - Toshiki Mabuchi

TI - An ℓ-th root of a test configuration of exponent ℓ

JO - Complex Manifolds

PY - 2016

VL - 3

IS - 1

SP - 169

EP - 185

AB - 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]).

LA - eng

KW - polarized algebraic manifold; desingularizations; test configurations

UR - http://eudml.org/doc/277104

ER -

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