发表机构
Kyushu University; Tohoku University（九州大学; 东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了容许正线丛的弱拟凸Oka流形满足Gromov椭圆性猜想，通过构造局部喷流并结合Oka逼近与粘合论证，还揭示了椭圆性在爆破及全纯族中的相关性质。
AI 中文摘要
我们证明，每一个容许正线丛的弱拟凸Oka流形都在Gromov意义下是椭圆的，从而证明了Gromov的椭圆性猜想。该证明建立了局部喷流的上同调构造：给定丛态射φ∶E→T_X，我们在E上引入二次φ-向量场，其流产生具有纤维导数φ的局部喷流，这类二次φ-向量场的存在性由E的对称化φ-阿蒂亚类的消失来刻画。这一结果在不附加任何Oka假设的情况下，为每一个容许正线丛的弱拟凸流形提供了局部支配喷流；而正性假设对局部存在性而言是必不可少的：代数维数为零的复环面在单点的爆破，以及代数维数为零的Kummer曲面，均不具有局部支配喷流。上述环面例子给出了紧致Kähler Oka流形，它们并非椭圆，表明椭圆性在爆破下不被保持，且在紧致Oka流形的全纯族中既非开也非闭。结合Xie与Zhao的工作，Kummer例子可得到非椭圆的Oka K3曲面。在附加Oka假设的情况下，我们将上述局部支配喷流整体化，该证明结合了Oka逼近与基于∂̄方程的加权L²估计的粘合论证。
英文摘要
We prove that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic in the sense of Gromov, thereby proving Gromov's ellipticity conjecture. The proof establishes a cohomological construction of local sprays. Given a bundle morphism $φ\colon E\to T_X$, we introduce quadratic $φ$-vector fields on $E$ whose flows yield local sprays with fibre derivative $φ$. Their existence is characterized by the vanishing of the symmetrized $φ$-Atiyah class of $E$. This gives a local dominating spray on every weakly pseudoconvex manifold admitting a positive line bundle, without any Oka assumption. The positivity hypothesis is essential even for local existence: blow-ups at a single point of complex tori of algebraic dimension zero, and Kummer surfaces of algebraic dimension zero, admit no local dominating spray. The torus examples give compact Kähler Oka manifolds that are not elliptic and show that ellipticity is not preserved under blowing up, and that it is neither open nor closed in holomorphic families of compact Oka manifolds. Combined with the work of Xie and Zhao, the Kummer examples yield Oka K3 surfaces that are not elliptic. Under the additional Oka assumption, we globalize the local dominating sprays constructed above. The proof combines Oka approximation with a gluing argument based on weighted $L^2$ estimates for the $\bar\partial$-equation.
Comments32 pages; comments welcome
