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Complex Manifolds

Ed. by Fino, Anna Maria


CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.643
Source Normalized Impact per Paper (SNIP) 2018: 0.812

Mathematical Citation Quotient (MCQ) 2017: 0.18

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2300-7443
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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:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-05-12 | DOI: https://doi.org/10.1515/coma-2016-0005

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 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]).

References

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  • [3] T. Fujita: On the structure of polarized varieties with Δ-genera zero, J. Fac. Sci., Univ. Tokyo, Sect. 1A, 22 (1975), 103–115. Google Scholar

  • [4] T. Mabuchi: A stronger concept of K-stability, a revised version of arXiv: math. DG 0910.4617, in preparation. Google Scholar

  • [5] T. Mabuchi: Test configurations with fixed components, in preparation. Google Scholar

  • [6] T. Mabuchi: The Yau-Tian-Donaldson conjecture for general polarizationns, II, in preparation. Google Scholar

  • [7] T. Mabuchi and Y. Nitta: Completion of the moduli space of test configurations, in preparation. Google Scholar

  • [8] E.H. Spanier: Algebraic Topology, MaGraw-Hill series in Higher Math., New York, 1966, 1–528. Google Scholar

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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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