On the full asymptotics of analytic torsion

The purpose of this article is to study the asymptotic expansion of Ray–Singer analytic torsion associated with powers p of a given positive line bundle over a compact n-dimensional complex manifold, as p→∞. Here we prove that the asymptotic expansion contains only the terms of the form pⁿ⁻ⁱlog⁡p,pⁿ⁻ⁱ for i∈N. For the first two leading terms it was proved by Bismut–Vasserot. We calculate the coefficients of the terms pⁿ⁻¹log⁡p,pⁿ⁻¹ in the Kähler case and thus answer the question posed in the recent work of Klevtsov–Ma–Marinescu–Wiegmann about quantum Hall effect. Our second result concerns the general asymptotic expansion of the analytic torsion for a compact complex orbifold.