AI 中文总结
研究拉回至万有覆盖上为恰当形式的凯勒形式的闭光滑流形，即辛非球面凯勒流形，探讨其拓扑几何性质，如大基本群、无正标量曲率凯勒度量，还研究相关猜想及复几何性质。
AI 中文摘要
我们对具有在万有覆盖上拉回为恰当形式的凯勒形式的闭光滑流形进行了详细研究。我们表明这些我们称为辛非球面凯勒流形的流形大量存在，甚至在非球面情形之外，并且具有有趣的拓扑和几何特征，比如像科拉尔那样的大基本群以及不存在正标量曲率的凯勒度量。受后者启发，我们在辛情形下针对黎曼度量解决了格罗莫夫 - Lawson 猜想的一个推广。我们还研究了辛非球面凯勒流形上的凯勒锥及其基本群的可实现性问题，并探索它们的其他复几何性质。
英文摘要
We present a detailed study of closed smooth manifolds having Kähler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical Kähler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group á la Kollár and the absence of Kähler metrics of positive scalar curvature. Motivated by the latter, we extend the Gromov--Lawson Conjecture on aspherical manifolds to symplectically aspherical manifolds and prove it in the spin case. We also study Kähler cones on symplectically aspherical Kähler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.
Commentsv3: Inaccuracies are corrected in Propositions 3.2 and 6.1, and acknowledgments are added. 24 pages, 1 figure
