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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615609
• 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 Kä hler 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 Kä hler world has been resolved by S.T. Yau et al. partly using our techniques. It says that a projective Kä hler X with H_X<0 has ample K_X. In the non-Kä hler 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 hyperkä hler 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。利用Nevanlinna理论、广义Ahlfors-Schwarz引理、交理论以及各种微分几何曲率条件之间的相互作用等，对曲线进行约束，得到了小林伪度量（即（反）双曲）的正性（分别为零）.我们还使用现代工具从代数几何在这项研究中，丰富的猜想在森的MMP是一个中心焦点。我们已经开始在这两个复和代数几何复兴，并旨在继续这条有前途的道路，进一步组织活动和促进HQP。 我们最近的研究集中在标准类K_X是nef的簇X上，包括那些没有有理曲线的簇和那些全纯截面曲率H_X是半负的簇。在得到了MMP中一类自然奇异簇的Bogomolov-Miyaoka-Yau不等式以及它们在相等情况下的一致化之后，我们的目标是寻找丰度猜想中最奇异的一类奇异簇。 G. Liu building on F. Zheng的工作证明了具有半负全纯双截曲率的射影Kähler流形被一个交换簇与一个具有充分K_X的簇的乘积所覆盖。我们的目标是同样的情况下，半负H_X和更一般的光滑品种X没有合理的曲线，通过我们的结果几乎阿贝尔纤维化，这将确认在这些各自的情况下丰富。一个希望的成分是这样一个簇X的平凡K_X被一个阿贝尔簇所覆盖，我们在半负H_X和aim的情况下证明了这一点。 S.小林证明了双曲簇X有充足的K_X.没有有理曲线的射影簇的类似物是Mori弯折定理。我们已经解决了模拟猜想的最佳奇异设置的dlt对，提供了一个几何版本的森锥定理在这个更一般的设置。我们还解决了在这种设置小林的猜想模上述希望的成分和丰度猜想，都知道三维。我们正在利用我们的新方法对线性系统的一般尖锐的结果。 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，当H_X<0时，射影Kähler X有充足的K_X。在非Kähler世界中，曲面结果已知模为一类VII曲面。后者已经看到了进步Apostolov和Dloussky，现在让我们研究他们的双曲通过类似的微分几何方法。 在我们的研究quasiAlbanese映射，我们有约束的全纯曲线的一般有限的情况下，并正在制定的代数情况。 我们得到了具有平凡K_X的超Kähler流形的伪度量为零，并且逼近于无穷小伪度量.