Nowhere-vanishing harmonic 1-forms on real loci of K3-fibred Calabi-Yau 3-folds

We develop an analytic construction of nowhere-vanishing harmonic $1$-forms on real loci of K3-fibred Calabi-Yau $3$-folds with collapsing Ricci-flat Kähler metrics.

We apply our construction to examples whose real loci have connected components diffeomorphic to $S^1\times S^2$ and to both trivial and nontrivial mapping tori.

As an application, we produce examples of compact $7$-manifold with holonomy $G_2$ via the Joyce-Karigiannis construction.