Limiting behavior of a class of Hermitian Yang-Mills metrics, I

Abstract

We begin to study the limiting behavior of Hermitian Yang-Mills metrics on a class of rank two slope-stable vector bundles over a product of two elliptic curves with respect to a family of Kähler metrics, which are flat and have areas ǫ and ǫ −1 on two elliptic curves respectively, approaching a large Kähler metric limit when ǫ → 0. The method involves the construction of a family of Hermitian metrics and comparison of these metrics with normalized Hermitian Yang-Mills metrics. We find these two families of metrics to be close, to arbitrary order in ǫ in any C k norm.

Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

“…Unfortunately we are unable to extend Lemma 6.43 from [ 18 ] to our setting, as our Poincare inequality (Proposition 5.2 ) requires a normalization that only holds fiberwise. Another result in this direction is proven by Fu in [ 22 ], who considers a specific rank two bundle over the product of two elliptic curves which is given by a two sheeted spectral cover. He defines a reference metric which satisfies desired asymptotic behavior near the ramification points of the cover, and then demonstrates that the a sequence of HYM metrics will converge smoothly to this reference metric.…”