Yang’s question on RC-positivity of tangent bundles

Yang’s question asks whether rational connectedness can be characterized by RC-positivity, or uniform RC-positivity, of the holomorphic tangent bundle. Earlier literature described this as Yang’s Problem 4.15 and left the converse unresolved.

Known results

• Yang’s results: uniform RC-positivity of $T_X$, or RC-positivity of every $\wedge^pT_X$, implies projectivity and rational connectedness.
• Yang, 2018: for projective rationally connected $X$, the dual tautological line bundle associated with $T_X$ is uniformly RC-positive, but this does not give uniform RC-positivity of $T_X$ itself.
• Qing, Sun, and Zhou, 2025: established partial converse results and described them as a step toward Yang’s question.
• A 2026 result: uniform weak RC-positivity of $T_X$ implies projectivity and rational connectedness, without proving the full converse.

September 2026 claimed equivalence

Chenghao Qing, Hongzhao Sun, and Xiangyu Zhou’s preprint Uniform RC-positivity of tangent bundles asserts equivalence among rational connectedness, RC-positivity, and uniform RC-positivity, with applications to blow-ups and selected non-Kähler surfaces. This would settle Yang’s question, but the version identified in the scan is an unrefereed version $1$ preprint.

Current status (as of September 2026): The full equivalence is claimed in an unrefereed preprint but is not independently verified; the earlier literature leaves the converse open.