The behavior of sequences of solutions to the Hitchin-Simpson equations

Abstract

The Hitchin-Simpson equations are first order non-linear equations for a pair of connection and a Higgs field, which is closely related to the Higgs bundle theory over Kähler manifold. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations over closed Kähler manifold with unbounded L 2 norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields.As an application, we study the realization problem of Taubes' Z2 harmonic 1-form over closed Kähler manifold. Using the Hitchin morphism for SL(2, C) Higgs bundle, we give a necessary and sufficient condition that whether a Z2 harmonic 1-form can be realized by a sequence of solutions to the Hitchin-Simpson equations.

The branched deformations of the special Lagrangian submanifolds

“…In this work, we'll look at a generalization of McLean's theorem to branched covering deformations using the Z 2 harmonic 1-forms, which could be thought of as a multivalued 1-forms satisfy extra constrains. The study of Z 2 harmonic 1-forms started from the work of Taubes [45] on characterized the non-compactness behavior of flat PSL(2, C) connections, see also [16,17,49] for different generalizations.…”