Abstract
Let be an ample bundle over a compact complex manifold. Fix a Hermitian metric in whose curvature defines a Kähler metric on . The Hessian of Mabuchi energy is a fourth-order elliptic operator on functions which arises in the study of scalar curvature. We quantize by the Hessian of balancing energy, a function appearing in the study of balanced embeddings. is defined on the space of Hermitian endomorphisms of endowed with the -inner product. We first prove that the leading order term in the asymptotic expansion of is . We next show that if is discrete, then the eigenvalues and eigenspaces of converge to those of . 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.
Citation
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
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