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

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 ${\pi }_{\ast }(L^k\otimes K_{\mathcal{X}/M})$ 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.

This preprint has been accepted by Geometriae Dedicata, and the final authenticated version will be available online at Springer Link https://link.springer.com/article/10.1007%2Fs10711-021-00625-y

References in corpus (3)

• On the full asymptotic of analytic torsion
• Geodesic-Einstein metrics and nonlinear stabilities
• Second variation of Selberg zeta functions and curvature asymptotics