Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation

Abstract

This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang--Mills metrics $H_{0,\epsilon}$ on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang--Mills metrics in the large K\"ahler limit. More precisely, the mean curvature of $H_{0,\epsilon}$ decays exponentially in every $C^k$-norm. Moreover, if $H_{1,\epsilon}$ denotes the exact Hermitian Yang--Mills metric and \[ H_\epsilon=H_{0,\epsilon}^{-1}H_{1,\epsilon}, \] then, after normalization, for every nonnegative integer $k$, there exist positive constants $C_k$ and $c_k$ such that \[ \|H_\epsilon- Id\|_{C^k}\leq C_k e^{-\frac{c_{k}}{\epsilon}}. \] The main analytic difficulty lies in the global $C^0$-comparison. Obtaining $C^0$-estimates for the coupled nonlinear Hermitian Yang--Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.