AI 中文总结
该研究证明卡拉比-丘流形在特定维数下不同胚于弱Fano簇，其基本群不同构于凯勒双曲流形的基本群，揭示了其与几何正性/负性类的拓扑刚性差异。
AI 中文摘要
我们研究卡拉比-丘流形的基础拓扑能否承载自然的几何正性或负性结构。在复偶维数 $n \geq 4$（当 $n \geq 6$ 时假设 $b_2=1$）的条件下，我们强化了Oguiso-Peternell定理，证明卡拉比-丘流形不同胚于弱Fano $n$ 维簇；该相同障碍也适用于具有拟正全纯截面曲率的凯勒流形。群作用类比尤其排除了带有孤立不动点的哈密顿圆作用的辛流形。在负性方面，我们证明卡拉比-丘流形的基本群不同构于凯勒双曲流形的基本群。综上，这些结果展现了一种共同的拓扑刚性，将卡拉比-丘流形与若干受几何正性或负性支配的基本类区分开来。
英文摘要
We study whether the topology underlying a Calabi-Yau manifold can support natural geometric positivity or negativity structures. In even complex dimension $n \geq 4$ (assuming $b_2=1$ when $n \geq 6$), we strengthen a theorem of Oguiso-Peternell by proving that a Calabi-Yau manifold is not homeomorphic to a weak Fano $n$-fold. The same obstruction applies to Kähler manifolds with quasi-positive holomorphic sectional curvature. A transformation-group analogue excludes, in particular, symplectic manifolds admitting Hamiltonian circle actions with isolated fixed points. On the negative side, we show that the fundamental group of a Calabi-Yau manifold is not isomorphic to that of a Kähler hyperbolic manifold. Taken together, these results exhibit a common topological rigidity separating Calabi-Yau manifolds from several fundamental classes governed by geometric positivity or negativity.
Comments18 pages
