  We investigate the algebraicity of compact K\"ahler manifolds admitting a
positive rational Hodge class of bidimension $(1,1)$. We prove that if the dual
K\"ahler cone of a compact K\"ahler manifold $X$ contains a rational class as
an interior point, then its Albanese variety is projective. As a consequence,
we answer the Oguiso--Peternell problem for Ricci-flat compact K\"ahler
manifolds. We also study related algebraicity problems for threefolds.
