Classification of Poor Manifolds in Low dimensions

Abstract

The notion of poor manifolds was introduced by Bandman and Zarhin, who asked for their classification. In this paper, we answer this question completely in dimensions at most 3. We also classify poor compact Kähler manifolds of arbitrary dimension under the additional assumption that Kodaira dimension is not $-\infty$. We classify all poor $K3$ surfaces. Finally, give a sufficient condition for a compact Kähle manifold to have Kodaira dimension greater than or equal to zero.