AI 中文总结
本文通过引入大二次曲面纤维化，结合bend-and-break论证与Hwang-Mok VMRT识别，证明了在切丛具有适配复结构条件下Blaschke猜想成立。
AI 中文摘要
我们提出了一种复几何方法来研究Blaschke猜想，即：一个其单射半径等于直径的流形等距于紧致秩一对称空间（CROSS）。特别地，我们引入了大二次曲面纤维化，它是大球面纤维化的复类比。要求其全空间为复子流形，同时解释了Blaschke流形预期具有的许多性质，包括其Clifford结构、Hopf纤维化和微分同胚类。此外，在标准CROSS情形下，大二次曲面丛与周围Fano簇的极小有理切线簇（VMRT）一致，我们通过一个在适当复化假设下的Blaschke猜想背景中的bend-and-break论证来解释这一点。将此与Hwang-Mok VMRT识别相结合，我们证明了对于其整个切丛上具有适配复结构的流形，Blaschke猜想成立。
英文摘要
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.
Comments21 pages, comments very welcome
