Fano流形上有理曲线的解析构造
Analytic Construction of Rational Curves on Fano Manifolds
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中文总结 AI 辅助
受Oka几何启发，利用Yau定理的正Ricci曲率度量引导全纯圆盘形变，构造Fano流形上的有理曲线，从而证明有理连通性与Hartshorne猜想。
中文摘要 AI 辅助
受Oka几何中构造整曲线方法的启发，我们给出了复Fano流形$X$上有理曲线的解析构造。Yau定理提供了具有正Ricci曲率的Kähler度量。利用该曲率引导全纯圆盘的形变，我们构造了半径趋于无穷且面积一致有界的圆盘映射。关键点在于在极限过程中保持导数归一化。这产生了一个有限面积的非平凡整映射$f:\mathbb C\rightarrow X$。该映射穿过无穷远点延拓为一个非平凡的全纯映射$\mathbb P^1\to X$。结合特征零的代数论证，该构造给出了Fano流形有理连通性以及Hartshorne关于丰沛切丛猜想的证明。
英文摘要
Inspired by constructions of entire curves in Oka geometry, we construct rational curves on complex Fano manifolds analytically, by alternately deforming a holomorphic disk to reduce its area and enlarging its source. Positive Ricci curvature yields an area-decreasing deformation, while an affine-lift perturbation enlarges the source at a controlled area cost. The two operations change the area and the weighted derivative by amounts we estimate explicitly, and analytic compactness passes from the resulting disks to a holomorphic sphere. The main result produces a sphere that meets a prescribed compact fiber without being contained in it, with an explicit area bound.
