Complex geometry of orbifold pairs and of their moduli spaces; structure, classification and relation to arithmetic geometry

轨道对及其模空间的复杂几何；

• 批准号: RGPIN-2022-05387
• 负责人: Lu, Steven
• 金额: $ 1.53万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759329
• 关键词: Complex; geometry; orbifold; pairs; their

We study algebraic varieties from the hyperbolicity perspective. We use various methods including Nevanlinna theory, Ahlfors-Schwarz lemmas, jets, etc, for constraining curves and for getting positivity of the Kobayashi pseudometric (i.e. hyperbolicity) and use modern algebraic geometry for getting its vanishing, the abundance conjecture being a central focus. More recently, we started also to look at the same for moduli spaces of varieties, specifically of manifolds with a fixed embedding into projective space. We have started a revival in this perpective in complex algebraic geometry and will propagate this promising path by organizing activities on them and by fostering of HQPs. Our recent focus centres on varieties whose canonical class K are nef, including varieties without rational curves and (by our results) varieties with seminegative holomorphic curvature. Having obtained the Bogomolov-Miyaoka-Yau inequality for singular varieties of klt type and their consequent uniformization in the case of equality, a project in our past proposal, we aim for the more singular lc case for a lead on the abundance problem. G. Liu building on F. Zheng's works showed that a projective Kähler manifold of seminegative holomorphic bisectional curvature is covered by a product of an abelian variety with an ample K variety. We aim for the same by dropping "bisectional", which would verify the abundance conjecture in this case, and more generally for smooth varieties without rational curves via our results on almost abelian fibrations. A hoped-for ingredient is that such a variety with trivial K be covered by an abelian variety, which we verified in the case of sem-inegative holomorphic curvature and aim in general. S. Kobayashi conjectured that a hyperbolic variety has ample K. The analog for a projective variety without rational curves is in essence Mori bend-and-break theorem. We have resolved the analog conjecture in the quasiprojective setting of log dlt pairs, providing a geometric version of Mori's cone theorem in this generalized setting. We have also resolved in this case Kobayashi's conjecture modulo the above hoped-for ingredient and the abundance conjecture, both known up to dimension three. We are exploiting new methods for these singular varieties for sharp results on linear systems. Kobayashi's conjecture in the Kähler world has been resolved by S.T. Yau et al. partially using our techniques. It says that a projective Kähler manifold of negative holomorphic curvature has ample K. In the non-Kähler world, the surface result is known modulo a class of VII0 surfaces for which we are investigating with the experts Apostolov and Dloussky. In our study of the quasiAlbanese map, we have constrained holomorphic curves for the generically finite case and are working out the algebraic case. We obtained the vanishing of the pseudometric for hyperkählers, manifolds with trivial K, and are rapidly closing in on the infinitesimal pseudometric.

我们从双曲线的角度研究代数品种。作为中心焦点。 HQP的促进，包括有理曲线的品种，（通过我们的结果）具有静态曲率的品种。 Zheng的作品的LC案例。几乎是Abelian Ferian ferian品种，我们的Verian品种覆盖了ABELIAN品种，这是Sem-Ins-Innegation Holomorphic Curvature的情况在log dlt对的准注射设置中的模拟猜想，在这种情况下，在这种情况下，他在莫里定理的几何版本中也解决了他的猜想这些对线性系统尖锐结果的新方法。 ，我们已经为一般有限的情况解释了全体形态曲线，并且正在计算代数情况。