Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation

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 $(M,\omega )$, let $\mathcal{G}=\mathrm{Aut}(P)$. For each $k\in \mathbb{N}$, we construct a $\mathcal{G}$-equivariant determinant line bundle ${\lambda }^{(k)}\to {\mathcal{J}}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of ${\lambda }^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed $2$-forms ${\Omega }_j$ on ${\mathcal{J}}_{int}$ and corresponding moment maps ${\mu }_j:{\mathcal{J}}_{int}\to C^{\infty }(M)$. Each ${\mu }_j$ arises from the asymptotic expansion of the variation of the logarithm of the Quillen metric with respect to Kähler potentials, with the complex structure held fixed. This provides a natural generalization of the Donaldson--Fujiki moment map interpretation of scalar curvature. Moreover, we show that ${\mu }_j$ coincide with the $Z$--critical equations introduced by Dervan--Hallam, and we state a generalization of Fujiki's fiber integral formula.

18 pages, final version. To appear in JSG