ChatGPT 5.6 Sol Pro Proves a Sharp Kähler Volume Comparison and Rigidity Theorem

In a preprint first uploaded on August 16, 2026, Ved Datar and Harish Seshadri present a sharp volume comparison for compact Kähler manifolds with positive holomorphic sectional curvature. If the holomorphic sectional curvature is at least $2$, the volume is at most $(2\pi)^n$, the volume of complex projective space with the corresponding Fubini-Study metric; equality holds exactly for that projective-space model.

The paper proves the stronger volume bound under a lower bound on a newly introduced mean RC curvature, a condition implied by both the stated holomorphic sectional-curvature bound and $\operatorname{Ric}\geq(n+1)\omega$. Datar and Seshadri say the proofs are due to ChatGPT 5.6 Sol Pro and describe the paper as an exposition of its output; they report verifying the proofs and accepting responsibility for errors.

The result is presented as solving the volume-comparison problem posed by Xiong and Yang without their earlier conjugate-radius assumption. Rigidity under the more general mean-RC-curvature hypothesis remains open: the paper proves rigidity in the holomorphic sectional-curvature and Ricci cases, but not for equality in the full mean-RC setting.