The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

TL;DR

This paper investigates the asymptotic behavior of the curvature of certain direct image bundles associated with high tensor powers of a relatively ample line bundle over a holomorphic fibration, revealing detailed curvature estimates and implications for analytic torsion.

Contribution

It provides the asymptotic expansion of the curvature of the $L^2$ and Quillen metrics on direct image bundles up to lower order terms than $k^{n-1}$ for large $k$, advancing understanding of geometric quantization.

Findings

Curvature asymptotics of direct image bundles are established.

The analytic torsion's logarithmic derivative grows slower than $k^{n-1}$.

Results apply to holomorphic fibrations with compact fibers and relatively ample line bundles.

Abstract

Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $\pi_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for large $k$. As an application we prove that the analytic torsion $\tau_k(\bar{\partial})$ satisfies $\partial\bar{\partial}\log(\tau_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is the dimension of fibers.