Classification of Poor Manifolds in Low dimensions

The notion of poor manifolds was introduced by Bandman and Zarhin, who asked for their classification. We study poor compact K\"ahler manifolds, i.e. those containing no rational curves and no codimension-one analytic subvarieties. We classify such manifolds in dimensions at most $3$, and in arbitrary dimension under the additional assumption that $\kappa(X)\neq -\infty$. We also describe the locus of poor $K3$ surfaces in the period domain.