Transgression on Hyperk\"{a}hler Manifolds and Generalized Higher Torsion Forms

Abstract

We propose a generalization of the Hodge $dd_c$-lemma to the case of hyperk\"ahler manifolds. As an application of this result we derive the global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk\"ahler manifolds. At the second part of the paper we consider the fourth order transgression for the infinite dimensional bundle arising from local families of hyperk\"ahler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for arising hypertorsion differential form. It's zero-degree part may be expressed in terms of the Laplace operators defined on the fibers of the local family.