par Fine, Joel 
Référence Duke Mathematical Journal, 161, 14, page (2753-2798)
Publication Publié, 2012
Référence Duke Mathematical Journal, 161, 14, page (2753-2798)
Publication Publié, 2012
Article révisé par les pairs
| Résumé : | Let L→X 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 D*D on functions which arises in the study of scalar curvature. We quantize D*D by the Hessian P*k Pk of balancing energy, a function appearing in the study of balanced embeddings. P*k Pk 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 P*k Pk is D*D. We next show that if Aut (X,L)/ℂ* is discrete, then the eigenvalues and eigenspaces of P*k Pk converge to those of D*D. 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. © 2012. |
