Blow-up preservation of positive holomorphic sectional curvature

Determine whether the blow-up along a subvariety of a projective manifold that admits a metric with positive holomorphic sectional curvature still carries a metric with positive holomorphic sectional curvature.

Background

The authors note that Yang proved compact Kähler manifolds with positive holomorphic sectional curvature are projective and rationally connected, resolving much of Yau’s conjecture. What remains is the question of whether positivity of holomorphic sectional curvature is preserved under blowing up along a subvariety.

This is explicitly identified as the remaining open part of Yau’s conjecture at the time of writing.