From Complex Hyperbolicity and Oka Geometry to Rational Curves on Fano Manifolds
数学专题报告
报告题目(Title)：From Complex Hyperbolicity and Oka Geometry to Rational Curves on Fano Manifolds
报告人(Speaker)：谢松晏 研究员（中科院）
地点(Place)：后主楼1125
时间(Time)：2026年11月17日（周二）14:00-15:00
邀请人(Inviter)：张科伟
报告摘要
Constructing rational curves on complex Fano manifolds by analytic methods has been a longstanding challenge. Our theorem that every rationally connected smooth complex projective manifold is Oka-1 suggests the analytic flexibility of Fano manifolds. It does not, however, give an independent construction of rational curves, since the known proofs of rational connectedness for Fano manifolds already use such curves.
I will explain how methods from Oka geometry and Brody's lemma lead to an analytic construction of rational curves. The key is an area-decreasing deformation of holomorphic discs, supplied by positive Ricci curvature, which allows further enlargement with uniformly bounded area and a fixed nonzero derivative at the center.
A nonconstant entire limit of finite area extends across infinity to a rational curve; a suitable rational curve may also appear earlier through bubbling. The guiding picture is the traditional Chinese technique of making baozi: stretching and gathering a dough wrapper before sealing it.
Combined with classical algebraic arguments, this construction gives new proofs in characteristic zero of the rational connectedness of Fano manifolds, due to Campana and Kollar-Miyaoka-Mori, and of Mori's solution of Hartshorne's conjecture on ample tangent bundles. The same method of deforming holomorphic discs also appears, without reference to our work, in a manuscript released by OpenAI on October 6, 2026, which presents a proof of Kobayashi's canonical-ampleness conjecture: every compact Kobayashi hyperbolic K\"ahler manifold $X$ has ample canonical bundle $K_X$.
This talk is based on joint work with Yun-Heng Du and Bin Guo.
