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Complex Manifolds

Ed. by Fino, Anna Maria

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

Toshiki Mabuchi
  • Corresponding author
  • Department of Mathematics, Osaka University, Toyonaka, Osaka, 560-0043, Japan
  • Other articles by this author:
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Published Online: 2016-05-12 | DOI: https://doi.org/10.1515/coma-2016-0005


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 on^X \^X 0, such that

whereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor on^X 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]).


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About the article

Received: 2015-10-10

Accepted: 2016-05-03

Published Online: 2016-05-12

Citation Information: Complex Manifolds, Volume 3, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2016-0005.

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© 2016 Toshiki Mabuchi. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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