Department of Mathematics, National Tsing Hua University

Abstract：The year 2026 has seen remarkable progress in AI-assisted mathematical research, including the disproof of the Erdős unit distance conjecture, advances related to the Riemann hypothesis, and proposed solutions to the complex-structure problem for the six-sphere and the Navier–Stokes problem. Inspired by these developments, we present an AI-assisted result in complex differential geometry: the holomorphic tangent bundle of any compact complex surface admitting a nonflat Ricci-flat Kähler metric carries an RC-positive Hermitian metric. As a consequence, we show that RC-positivity is not preserved under taking tensor, exterior, or symmetric powers. Moreover, RC positivity is strictly weaker than uniform RC-positivity, and RC-positivity of the holomorphic tangent bundle does not necessarily imply rational connectedness of the underlying manifold. This is joint work with You-Cheng Chow.