Weyl structures with special holonomy on compact conformal manifolds

We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl structure $\mathrm{\nabla }$, i.e. a torsion-free connection preserving the conformal structure, which is locally but not globally the Levi-Civita connection of a metric in $[g]$. Our aim is to classify all such structures when both $\mathrm{\nabla }$ and ${\mathrm{\nabla }}^g$, the Levi-Civita connection of $g$, have special holonomy. In such a setting, $(M,[g],\mathrm{\nabla })$ is either flat, or irreducible, or carries a locally conformally product (LCP) structure. Since the flat case is already completely classified, we focus on the last two cases. When $\mathrm{\nabla }$ has irreducible holonomy we prove that $(M,g)$ is either Vaisman, or a mapping torus of an isometry of a compact nearly Kähler or nearly parallel ${\mathrm{G}}_2$ manifold, while in the LCP case we prove that $g$ is neither Kähler nor Einstein, thus reducible by the Berger-Simons Theorem, and we obtain the local classification of such structures in terms of adapted metrics.