  We prove the vanishing of the first Betti number on compact manifolds
admitting a Weyl structure whose Ricci tensor satisfies certain positivity
conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry.
We also study compact Hermitian-Weyl manifolds with non-negative symmetric part
of the Ricci tensor of the canonical Weyl connection and show that every such
manifold has first Betti number $b_1 =1$ and Hodge numbers $h^{p,0} =0$ for
$p>0$, $h^{0,1} =1$, $h^{0,q} =0$ for $q>1$.
