Blaschke Conjecture and Complex Geometry

Abstract

We provide a complex-geometric approach to the Blaschke conjecture, i.e., that a manifold whose injectivity radius equals its diameter is isometric to a compact rank-one symmetric space (CROSS). In particular, we introduce the great quadric fibration, a complex analogue of the great sphere fibration. Requiring its total space to be a complex submanifold simultaneously explains many properties that a Blaschke manifold is expected to possess, including its Clifford structure, Hopf fibration, and diffeomorphism class. Furthermore, the great quadric bundle coincides with the variety of minimal rational tangents (VMRT) of the ambient Fano variety in the standard CROSS cases, and we explain this through a bend-and-break argument in the setting of the Blaschke conjecture under a suitable complexification assumption. Combining this with Hwang-Mok VMRT recognition, we prove the Blaschke conjecture for manifolds admitting an adapted complex structure on their entire tangent bundles.