Classify compact positive-scalar-curvature Kähler manifolds in higher dimensions

Classify compact positive scalar curvature Kähler manifolds in complex dimension at least three.

Background

The paper explains that positive scalar curvature Kähler metrics impose strong geometric restrictions, including uniruledness, and recalls a classification for compact complex surfaces of Kähler type. Beyond complex dimension two, however, the global geometry of compact manifolds admitting such metrics is not determined. A classification in complex dimension at least three would clarify which complex-geometric structures permit positive scalar curvature Kähler metrics.

References

To our knowledge, no classification of compact PSC Kähler manifolds is known in complex dimension at least three.

— A Variational Characterization of Positive Scalar Curvature Kähler metrics  (2608.20023 - Sha, 20 Aug 2026) in Section 1.1, Introduction

A general geometric characterization of the manifolds satisfying \mathcal K_X{\mathrm{psc}}=\mathcal T_X+ remains open.

— A Variational Characterization of Positive Scalar Curvature Kähler metrics  (2608.20023 - Sha, 20 Aug 2026) in Question 1.1, Section 1.3, Criteria for positive scalar curvature Kähler metrics