Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation
Kiyoon Eum
TL;DR
The paper develops a global, asymptotic determinant-line-bundle framework to interpret higher-order curvature data in Kähler geometry as moment maps. By constructing a family of $\,\mathcal{G}$-equivariant determinant line bundles $\\lambda^{(k)}$ with Quillen metrics over the space $\\mathcal{J}_{int}$, it derives a hierarchy of invariant two-forms $\\Omega_j$ and corresponding moment maps $\\mu_j$ from the $k\to\infty$ expansion of the log-Quillen metric variation with respect to Kähler potentials. The main technical contributions connect these $\\mu_j$ to $Z$-critical equations with central charge determined by the Todd class and establish a generalized Fujiki fiber integral formula, recovering the classical scalar-curvature case at $j=1$ and aligning with Foth–Uribe in the appropriate limit. The results provide a unifying determinant-line-bundle perspective on stability notions in Kähler geometry and offer a new route to studying canonical metrics via equivariant index theory and Toeplitz quantization.
Abstract
In Kähler geometry, the Donaldson-Fujiki moment map picture interprets the scalar curvature of a Kähler metric as a moment map on the space of compatible almost complex structures on a fixed symplectic manifold. In this paper, we generalize this picture using the framework of equivariant determinant line bundles. Given a prequantization $P=(L,h,\nabla)$ of a compact symplectic manifold $(M,ω)$, let $\mathcal{G}=\mathrm{Aut}(P)$. We construct for each $k\in\mathbb{N}$ a $\mathcal{G}$-equivariant determinant line bundle $λ^{(k)}\rightarrow\mathcal{J}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of $λ^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed two-forms $Ω_j$ on $\mathcal{J}_{int}$ and corresponding moment maps $μ_j:\mathcal{J}_{int}\rightarrow C^\infty(M)$. Each $μ_j$ arises from the asymptotic expansion of the variation of the log of the Quillen metric with respect to Kähler potentials, keeping the complex structure fixed. This provides a natural generalization of the Donaldson-Fujiki moment map interpretation of scalar curvature. Moreover, we show that $μ_j$ coincide with the $Z$-critical equations introduced by Dervan-Hallam, and we state a generalization of Fujiki's fiber integral formula.