Yau’s conjecture on positive holomorphic sectional curvature

Determine whether a compact Kähler manifold with positive holomorphic sectional curvature is unirational and whether it is projective; furthermore, determine whether a projective manifold obtained by blowing up a compact manifold with positive holomorphic sectional curvature along a subvariety still admits a metric with positive holomorphic sectional curvature.

Background

Holomorphic sectional curvature is a central invariant in complex differential geometry. Yau posed a well-known conjecture asking, among other things, whether positive holomorphic sectional curvature implies projectivity or even unirationality, and whether positivity persists after blowing up along a subvariety.

The paper recalls this conjecture to motivate applications of Hermitian tensor techniques to curvature questions. Subsequent text notes partial progress (e.g., projectivity is known), but the full conjecture is stated explicitly here as context.