Hartshorne’s conjecture on ample tangent bundles

Hartshorne’s conjecture says that for a smooth projective variety $X$, an ample tangent bundle characterizes projective space: $X \cong \mathbb{P}^n$. Mori established this classification in 1979.

Known results

• Mori, 1979: $T_X$ is ample exactly when $X \cong \mathbb{P}^n$ for smooth projective $X$.
• Sieder, 2017: extended the characterization to normal projective varieties with ample tangent sheaf.

September 2026 analytic construction

Yun-Heng Du, Bin Guo, and Song-Yan Xie claim that finite-area entire maps from the plane produce rational curves implying the same classification consequence. The newly posted preprint’s claimed route has not been independently verified.

Current status (as of September 2026): Mori’s classification is settled; the new analytic argument is an unverified claimed alternative rather than an established change in status.