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.

我们从双曲性的角度研究代数簇。我们使用各种方法，包括Nevanlinna理论，Ahlfors-Schwarz引理，喷气机等，约束曲线和获得积极的小林伪度量（即双曲），并使用现代代数几何得到其消失，丰度猜想是一个中心焦点。最近，我们也开始研究簇的模空间，特别是在射影空间中有固定嵌入的流形。我们已经开始在复代数几何的这一视角的复兴，并将通过组织活动和培养HQP来传播这条有前途的道路。我们最近的焦点集中在品种的典型类K是nef，包括品种没有合理的曲线和（我们的结果）品种半负全纯曲率。在获得了Bogomolov-Miyaoka-Yau不等式的奇异品种的klt型和随之而来的均匀化的情况下，平等，在我们过去的建议，我们的目标是更奇异的LC情况下，导致的丰度问题。G. Liu building on F.郑的作品表明，一个投射凯勒流形的半负全纯双截曲率所涵盖的产品的阿贝尔品种丰富的K品种。我们的目标是相同的下降“bisectional”，这将验证在这种情况下的丰度猜想，更一般的光滑品种没有合理的曲线，通过我们的结果几乎阿贝尔纤维。一个希望的成分是，这样的一个品种平凡K覆盖的阿贝尔品种，我们验证的情况下，半负的全纯曲率和目标一般。S.小林指出双曲簇有足够的K。没有有理曲线的射影簇的类比本质上是Mori弯折定理。我们已经解决了类似的猜想中的拟投射设置的日志dlt对，提供了一个几何版本的森锥定理在这个广义的设置。在这种情况下，我们也解决了小林猜想模上述希望的成分和丰度猜想，都知道到三维。我们正在开拓新的方法，这些奇异品种的尖锐的结果线性系统。小林在Kähler世界中的猜想已被S. T. Yau等人部分使用了我们的技术。它说一个具有负全纯曲率的射影Kähler流形有充足的K。在非凯勒世界，表面的结果是已知模一类VII 0表面，我们正在调查与专家Apostolov和Dloussky。在拟Albanese映射的研究中，我们已经得到了一般有限情形下的约束全纯曲线，并且正在研究代数情形。我们得到了超kählers流形（具有平凡K的流形）的伪度量的消失，并且正在迅速接近无穷小的伪度量。