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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=658030
• 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<0 has ample K_X. In the non-Kähler world, the surface result is known modulo a class of VII surfaces. The latter has seen advances by Apostolov and Dloussky that now allow us to study their hyperbolicity via similar differential geometric methods.******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 hyperkähler manifolds, which have trivial K_X, and are closing in on the infinitesimal pseudometric.**

我们从双曲性的角度研究代数簇，我们使用 Nevanlinna 理论、广义 Ahlfors-Schwarz 引理、交集理论以及各种微分几何曲率条件之间的相互作用等来约束曲线并获得小林伪度量的正性（分别消失）。 （即（反）双曲性）。在这项研究中，我们还使用了代数几何的现代工具，即丰度猜想。 Mori 的 MMP 是一个中心焦点。我们已经开始在复杂几何和代数几何方面复兴，并旨在通过进一步组织活动和培养 HQP 来继续这条有前途的道路。 ****** 我们最近的重点集中在品种 X 上。其规范类 K_X 为 nef，包括那些没有有理曲线的类和全纯截面曲率 H_X 为半负的类。 获得了奇异簇自然类的 Bogomolov-Miyaoka-Yau 不等式。在MMP及其随后在相等的情况下的统一化中，我们的目标是为丰度猜想找到最奇异的此类。******G.刘在F.Zheng的作品的基础上表明了半负全纯的射影凯勒流形二分曲率由具有充足 K_X 的阿贝尔簇的乘积覆盖，我们的目标是对于半负 H_X 的情况，更一般地对于没有有理数的平滑簇 X。通过我们对几乎阿贝尔纤维的结果得出的曲线，这将证实这些各自情况下的丰度，一个希望的成分是这样的具有微不足道的 K_X 的变量 X 被阿贝尔变量覆盖，我们在半负 H_X 和目标的情况下验证了这一点。一般而言，小林猜想，双曲簇 X 具有充足的 K_X，而无理曲线的射影簇的模拟是 Mori 弯曲和断裂定理。 dlt 对的最佳奇异设置中的猜想，在这个更一般的设置中提供了森圆锥定理的几何版本，我们还在这个设置中解决了小林猜想对上述期望成分和丰度猜想的模数，两者都已知到维度。 3.我们正在利用我们的新方法来获得线性系统上的一般锐结果。*****小林在凯勒世界中的猜想已被 S.T Yau 等人解决。部分使用我们的技术，它表示 H_X<0 的投影 Kähler X 具有充足的 K_X 在非 Kähler 世界中，曲面结果以 VII 类曲面为模，现在已经看到了 Apostolov 和 Dloussky 的进展。允许我们通过类似的微分几何方法来研究它们的双曲性。********在我们对拟阿尔巴尼亚映射的研究中，我们对一般有限情况有约束全纯曲线，并且是计算出代数情况。*****我们获得了超卡勒流形的伪度量消失，它具有微不足道的 K_X，并且接近无穷小伪度量。**