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Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group Read chapter Read first chapter

2016 | OriginalPaper | Chapter

The Yau-Tian-Donaldson Conjecture for General Polarizations, I

Author : Toshiki Mabuchi

Published in: Geometry and Topology of Manifolds

Publisher: Springer Japan

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Abstract

In this paper, some \(C^0\) boundedness property (BP) is introduced for balanced metrics on a polarized algebraic manifold (X, L). Then by assuming that (X, L) is strongly K-stable in the sense of [8], we shall show that the balanced metrics have (BP). In a subsequent paper [10], this property (BP) plays a very important role in the study of the Yau-Tian-Donaldson conjecture for general polarizations.

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previous chapter Willmore 2-Spheres in : A Survey
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Footnotes
1
In [13], we define some algebraic concept of “real K-stability” which will be shown to be equivalent to strong K-stability. By using this, we shall show in [15] that the definition of strong K-stability is independent of the choice of h.
 
2
More precisely, the original Donaldson-Futaki invariant and our definition differ only by multiplication by \(L^1\)-norm of the test configuration.
 
3
In our approach using balanced metrics, nonsingularity of X is of crucial importance. If X were a polarized algebraic singular orbifold of complex dimension 2 only with \(A^1\)-singularities, then X could admit no balanced metrics, since the orbifold metric obtained as the pullback to X of the Fubini-study metric would vanish at the singular points.
 
4
By \(\exp \{-2{\text {dist}}(h, h_{\ell })\} \le \{\varSigma _{\alpha =1}^{N_{\ell }} |\sigma _{\ell , \alpha }|^2\}^{1/\ell } \{\varSigma _{\alpha =1}^{N_{\ell }} |\tau _{\ell , \alpha }|^2\}^{-1/\ell } \le \exp \{2{\text {dist}}(h, h_{\ell })\}\), this makes sense, since \(h_{\ell }\) coincides with \(\{\varSigma _{\alpha =1}^{N_{\ell }} |\sigma _{\ell , \alpha }|^2\}^{-1/\ell }\) up to a constant multiple, and since by [19], \(\{\varSigma _{\alpha =1}^{N_{\ell }} |\tau _{\ell , \alpha }|^2\}^{-1/\ell }\) approximates h, up to a constant multiple, in \(C^{\infty }\) as \(\ell \rightarrow \infty \).
 
Literature
1.
Donaldson, S.K.: Scalar curvature and projective embeddings. I. J. Differ. Geom. 59, 479–522 (2001)MathSciNetMATH
2.
Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)MathSciNetMATH
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7.
Mabuchi, T.: Asymptotics of polybalanced metrics under relative stability constraints. Osaka J. Math. 48, 845–856 (2011)
8.
Mabuchi, T.: The Donaldson-Futaki invariant for sequences of test configurations. In: Geometry and Analysis on Manifolds, Progress in Mathematics, vol. 308, pp. 395–403. Birkhäuser, Boston (2015)
9.
10.
Mabuchi, T.: The Yau-Tian-Donaldson conjecture for general polarizations, II (in preparation)
11.
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13.
Mabuchi, T.: Test configurations with fixed components (in preparation)
14.
Mabuchi, T., Nitta, Y.: Strong K-stability and asymptotic Chow stability. In: Geometry and Analysis on Manifolds, Progress in Mathematics, vol. 308, pp. 405–411. Birkhäuser Boston (2015)
15.
Mabuchi, T., Nitta, Y.: Completion of the moduli space of test configurations (in preparation)
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Metadata
Title
The Yau-Tian-Donaldson Conjecture for General Polarizations, I
Author
Toshiki Mabuchi
Copyright Year
2016
Publisher
Springer Japan
Book
Geometry and Topology of Manifolds

Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-4-431-56021-0_13

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