Quantisation and the Hessian of Mabuchi energy

TL;DR

This paper studies the quantization of the Hessian of Mabuchi energy on complex manifolds, showing convergence of eigenvalues and eigenspaces, with implications for scalar curvature and Kähler metrics.

Contribution

It establishes the asymptotic relation between the Hessian of balancing energy and the Mabuchi Hessian, including eigenvalue convergence and applications to scalar curvature.

Findings

Asymptotic expansion of E(k) matches D at leading order.

Eigenvalues and eigenspaces of E(k) converge to those of D.

Results imply sharpness of Phong-Sturm estimate and answer Donaldson's question.

Abstract

Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a K\"ahler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a function appearing in the study of balanced embeddings. E(k) is defined on the space of Hermitian endomorphisms of H^0(X, L^k), endowed with the L^2-innerproduct. We first prove that the leading order term in the asymptotic expansion of E(k) is D. We next show that if Aut(X,L) is discrete modulo scalars, then the eigenvalues and eigenspaces of E(k) converge to those of D. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature K\"ahler metric. As consequences of our results we prove that a certain estimate of Phong-Sturm is sharp and give a negative answer to a question of Donaldson. We also discuss some possible applications to the study of Calabi flow.