Some aspects of positivity in complex geometry

复杂几何中积极性的某些方面

• 批准号: 254956526
• 负责人: Professor Dr. Thomas Peternell
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/254956526?language=en
• 关键词: Some; aspects; positivity; complex; geometry

Manifolds with semi-positive Ricci curvature are central objects in complex geometry. In many aspects, this class however is too small. Instead, one considers varieties admitting a singular metric with semi-positive curvature current. From an algebraic point of view, this is a property which need only to be checked along curves. Technically, we speak of varieties with nef anticanonical classes. This varieties are in the focus of the project. In particular, we aim to study the global structure of those manifolds (Albanese, birational geometry, algebraic reduction). We further want to investigate, whether a manifold with nef anticanonical class can be deformed into varieties with semi-positive Ricci curvature. These subclass is much better understood, using differential-geometric methods. Moreover, we plan to investigate whether general non-algebraic Kähler manifolds with nef anticanonical classes can be approximated by algebraic ones. This would allow to use algebraic methods to study these Kähler varieties. Finally, general positivity notions for line bundles and their connections to the geometry of curves with positive normal bundles with be studied in the frame of a duality theory.

具有半正Ricci曲率的流形是复几何中的中心对象。但在很多方面，这类人太少了。相反，考虑品种承认一个奇异度量与半正曲率电流。从代数的观点来看，这是一个只需要沿沿着曲线检查的性质。从技术上讲，我们说的品种与nef anticanonical类。这些品种是该项目的重点。特别是，我们的目标是研究这些流形的整体结构（阿尔巴尼亚，双有理几何，代数约化）。我们还想进一步研究一个具有nef反正则类的流形是否可以变形为具有半正Ricci曲率的簇。这些子类是更好地理解，使用微分几何方法。此外，我们计划调查是否一般的非代数Kähler流形nef anticanonical类可以近似代数。这将允许使用代数方法来研究这些凯勒品种。 最后，在对偶理论的框架下研究了线丛的一般正性概念及其与具有正法丛的曲线几何的联系。