The Hitchin-Simpson equations defined over a Kaehler manifold are first order, non-linear equations for a pair of a connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin–Simpson equations with norms of these 1-forms unbounded. We will talk about two applications of this compactness theorem, one is the realization problem of the Taubes’ Z2 harmonic 1-form and another is the Hitchin’s WKB problem in higher dimensional. We will also discuss some open questions related to this question.

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Last updated: 11 Dec 2023