Vanishing Theorems for Locally Conformal Hyperkähler Manifolds

Let $M$ be a compact locally conformal hyperkähler manifold. We prove a version of the Kodaira–Nakano vanishing theorem for $M$. This is used to show that $M$ admits no holomorphic differential forms and the cohomology of the structure sheaf $H^i(\mathcal O_M)$ vanishes for $i>1$. We also prove that the first Betti number of $M$ is $1$. This leads to a structure theorem for locally conformal hyperkähler manifolds that describes them in terms of $3$-Sasakian geometry. Similar results are proven for compact Einstein–Weyl locally conformal Kähler manifolds.