On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
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Tomoyuki Hisamoto
Abstract
We apply our integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt Nyström to the effect that the sequence of spectral measures for the induced ℂ*-action on the central fiber converges to the canonical measure defined by the associated weak geodesic ray in the space of Kähler metrics. This limit measure coincides with the classical Duistermaat–Heckmann measure if the test configuration is product. As a consequence, we show that the algebraic p-norm of the test configuration is equal to the Lp-norm of tangent vectors on the geodesic ray. Using this result, we give a natural energy theoretic explanation for the lower bound estimate on the Calabi functional by Donaldson, extending the statement to any p-norm (p ≥ 1), and prove an analogous result for Kähler–Einstein metrics.
Funding source: JSPS Research Fellowships for Young Scientists
Award Identifier / Grant number: 22-6742
The author would like to express his gratitude to his advisor Professor Shigeharu Takayama for his warm encouragements, suggestions and reading the drafts. The author also would like to thank Professor Sébastien Boucksom for his indicating the relation between the author's paper [Math. Z. 275 (2013), no. 1–2, 233–243] and the paper of Julius Ross and David Witt Nyström. The author wishes to thank Doctor David Witt Nyström, Professor Robert Berman and Professor Bo Berndtsson for stimulating discussion and helpful comments during the author's stay in Gothenburg. In particular, the formulation of Theorem 1.1 and Theorem 1.2 are due to the suggestions of Doctor David Witt Nyström. The proof of Theorem 1.1 is also indebted to his many helpful comments. Our work was carried out at several institutions including Charmers University of Technology, Gothenburg University and University of Tokyo. We gratefully acknowledge their support. I wish to thank the anonymous referees for their many helpful comments, especially for the indication that using Lemma 1.6 of [Compos. Math. 147 (2011), no. 4, 1205–1229] one can drop the assumption 𝒳 is normal. The situation is also explained in Remark 4.1.
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Congruence kernels around affine curves
- On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
- On the splitting of quasilinear p-forms
- Graded quiver varieties and derived categories
- On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
- Static Klein–Gordon–Maxwell–Proca systems in 4-dimensional closed manifolds. II
- Linear stability of algebraic Ricci solitons
- Dynamical stability of algebraic Ricci solitons
- Erratum to Dynamics of the Krichever construction in several variables (J. reine angew. Math. 572 (2004), 111–138)
- Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)
Artikel in diesem Heft
- Frontmatter
- Congruence kernels around affine curves
- On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
- On the splitting of quasilinear p-forms
- Graded quiver varieties and derived categories
- On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
- Static Klein–Gordon–Maxwell–Proca systems in 4-dimensional closed manifolds. II
- Linear stability of algebraic Ricci solitons
- Dynamical stability of algebraic Ricci solitons
- Erratum to Dynamics of the Krichever construction in several variables (J. reine angew. Math. 572 (2004), 111–138)
- Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)
