The second variation of analytic torsion on teichmüller space

Journal article, 2025

Let pi : X -> T be the Teichm & uuml;ller curve over the Teichm & uuml;ller space T, where each fiber Xy is the Riemann surface of genus g >= 2 corresponding to the point y is an element of T. In this paper, we study the direct image bundle E-k = pi(& lowast;)(L-k circle times K-X/T), naturally equipped with an L-2-metric, where L is a relatively ample line bundle over X and restricts to a rational power of the canonical bundle on each fiber. We analyze the asymptotic behavior of the curvature of the induced L-2-metric on the determinant line bundle det E-k as k ->infinity. As a consequence, we show that the second variation of the logarithmic analytic torsion log tau(k)(partial derivative) decays faster than any polynomial rate.