Quantization and the Hessian of Mabuchi energy

Abstract

Let $L\to X$ 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 ${\mathcal{D}}^{\ast }\mathcal{D}$ on functions which arises in the study of scalar curvature. We quantize ${\mathcal{D}}^{\ast }\mathcal{D}$ by the Hessian $P_k^{\ast }{P}_k$ of balancing energy, a function appearing in the study of balanced embeddings. $P_k^{\ast }{P}_k$ is defined on the space of Hermitian endomorphisms of $H^0(X,L^k)$ endowed with the $L^2$-inner product. We first prove that the leading order term in the asymptotic expansion of $P_k^{\ast }{P}_k$ is ${\mathcal{D}}^{\ast }\mathcal{D}$. We next show that if $Aut(X,L)/{\mathbb{C}}^{\ast }$ is discrete, then the eigenvalues and eigenspaces of $P_k^{\ast }{P}_k$ converge to those of ${\mathcal{D}}^{\ast }\mathcal{D}$. 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.