Generalized hyperbolicity and the geometry of algebraic varieties

广义双曲性和代数簇的几何

• 批准号: RGPIN-2016-05294
• 负责人: Lu, Steven
• 金额: $ 1.6万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=709142
• 关键词: Generalized; hyperbolicity; geometry; algebraic; varieties

We study algebraic varieties from the hyperbolicity perspective. We use Nevanlinna theory, Generalized Ahlfors-Schwarz lemmas, intersection theory and the interplay with and between various differential geometric curvature conditions, etc, for constraining curves and getting positivity (respectively vanishing) of the Kobayashi pseudometric (i.e. (anti)hyperbolicity). We also use modern tools from algebraic geometry in this study, the abundance conjecture in Mori's MMP being a central focus. We have started a revival in this both in complex and in algebraic geometry and aim to continue this promising path by further organized activities and fostering of HQPs. Our recent focus centres on varieties X whose canonical class K_X are nef, including those without rational curves and those whose holomorphic sectional curvature H_X is seminegative. Having obtained the Bogomolov-Miyaoka-Yau inequality for a natural class of singular varieties in the MMP and their consequent uniformization in the case of equality, we aim for the most singular such class for the abundance conjecture. 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 a variety having ample K_X. We aim for the same for the case of seminegative H_X and more generally for smooth varieties X without rational curves via our results on almost abelian fibrations, which would confirm abundance in these respective cases. A hoped-for ingredient is that such a variety X with trivial K_X be covered by an abelian variety, which we verified in the case of seminegative H_X and aim in general. S. Kobayashi conjectured that a hyperbolic variety X has ample K_X. The analog for a projective variety without rational curves is Mori bend-and-break theorem. We have resolved the analog conjecture in the optimal singular setting of dlt pairs, providing a geometric version of Mori's cone theorem in this more general setting. We have also resolved in this setting Kobayashi's conjecture modulo the above hoped-for ingredient and the abundance conjecture, both known up to dimension three. We are exploiting our new methods for general sharp results on linear systems. Kobayashi's conjecture in the Kähler world has been resolved by S.T. Yau et al. partly using our techniques. It says that a projective Kähler X with H_X

我们从双曲线的角度研究代数。在这项研究中。我们已经在复杂和代数几何形状中开始复兴，并旨在通过进一步的有组织的活动和HQP的促进来继续这一有希望的道路。 我们最近的重点是X的X型X，其曲率为NEF曲率H_X是半指的。大多数奇异的阶级是丰富的猜想。 G. liu在F. Zhed上的建筑物，半塑形双形曲率的投影kähler被覆盖的As覆盖的As覆盖，其中一些Abelian品种具有良好的K_X。对于平滑品种x而在不合理曲线的情况下，通过我们的几乎阿贝尔纤维的结果，希望成分的成分是，与abelian品种一起覆盖了一种带有琐碎k_x的品种的工作，我们在半凝H_x的情况下进行了验证，并在一般情况下进行瞄准并瞄准。 。 S. kobayashi猜想，双曲线品种具有足够的k_x。一般的结束和丰富的猜想，均已为第三维度。 KählerWorld的Kobayashi猜想是S.Au等人的解决方案