Classification of Poor Manifolds in Low dimensions

TL;DR

This paper classifies poor compact Kähler manifolds in low dimensions and under certain conditions, providing insights into their structure and the locus of poor K3 surfaces within the period domain.

Contribution

It offers a classification of poor manifolds in dimensions up to 3 and extends results to higher dimensions with specific assumptions, also describing poor K3 surfaces in the period domain.

Findings

Classified poor manifolds in dimensions ≤3.

Extended classification to higher dimensions with κ(X) ≠ -∞.

Described the locus of poor K3 surfaces in the period domain.

Abstract

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.