1 November 2012 Quantization and the Hessian of Mabuchi energy
Joel Fine
Duke Math. J. 161(14): 2753-2798 (1 November 2012). DOI: 10.1215/00127094-1813524


Let LX be an ample bundle over a compact complex manifold. Fix a Hermitian metric in L whose curvature defines a Kähler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator DD on functions which arises in the study of scalar curvature. We quantize DD by the Hessian PkPk of balancing energy, a function appearing in the study of balanced embeddings. PkPk is defined on the space of Hermitian endomorphisms of H0(X,Lk) endowed with the L2-inner product. We first prove that the leading order term in the asymptotic expansion of PkPk is DD. We next show that if Aut(X,L)/C is discrete, then the eigenvalues and eigenspaces of PkPk converge to those of DD. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature Kähler metric. As consequences of our results we prove that an estimate of Phong and Sturm is sharp and give a negative answer to a question posed by Donaldson. We also discuss some possible applications to the study of Calabi flow.


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Joel Fine. "Quantization and the Hessian of Mabuchi energy." Duke Math. J. 161 (14) 2753 - 2798, 1 November 2012. https://doi.org/10.1215/00127094-1813524


Published: 1 November 2012
First available in Project Euclid: 26 October 2012

zbMATH: 1262.32024
MathSciNet: MR2993140
Digital Object Identifier: 10.1215/00127094-1813524

Primary: 32Q15
Secondary: 53D50

Rights: Copyright © 2012 Duke University Press


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Vol.161 • No. 14 • 1 November 2012
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