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.**

我们从双曲的角度研究代数变分。利用Nevanlinna理论、广义Ahlfors-Schwarz引理、交点理论以及与各种微分几何曲率条件的相互作用等，对曲线进行了约束，得到了Kobayashi伪度量（即（反）双曲）的正性（分别消失）。在这项研究中，我们还使用了代数几何中的现代工具，在Mori的MMP中，丰度猜想是一个中心焦点。我们已经开始在复杂几何和代数几何中复兴，并打算通过进一步组织活动和培养hqp来继续这条有希望的道路。******我们最近的焦点集中在正则类K_X为nef的变种X上，包括那些没有有理曲线的变种和那些全纯截面曲率H_X为半负的变种X。在得到了MMP中奇异变异的自然类的Bogomolov-Miyaoka-Yau不等式及其在相等情况下的均匀化之后，我们的目标是求丰度猜想的最奇异类。******G。Liu在郑F.的基础上证明了半负全纯对分曲率的射影Kähler流形被一个具有充足K_X的阿贝尔变体的积所覆盖。我们的目标是在半负H_X的情况下，通过我们在几乎阿贝尔振动上的结果，对于没有有理曲线的光滑品种X，我们的目标是相同的，这将证实在这些各自的情况下的丰度。一个期望的成分是这样一个具有平凡K_X的变种X被一个阿贝尔变种所覆盖，我们在半负H_X的情况下验证了这一点，并将其作为一般的目标。* * * * * *。Kobayashi推测，双曲型X有充足的K_X。无有理曲线的射影变化的类比是Mori弯曲-断裂定理。我们解决了dlt对的最优奇异设置中的模拟猜想，在这种更一般的设置中提供了Mori锥定理的几何版本。在这种情况下，我们还解决了小林猜想对上述期望成分的模量和丰度猜想，这两个猜想都是在三维空间中已知的。我们正在利用我们的新方法在线性系统上得到一般的尖锐结果。******Kobayashi在Kähler世界的猜想已经被S.T. Yau等人部分地利用我们的技术解决了。它表示H_X<0的射影Kähler X有充足的K_X。在non-Kähler世界中，曲面结果对一类VII曲面取模是已知的。后者已经看到阿波斯托洛夫和德劳斯基的进步，现在允许我们通过类似的微分几何方法来研究它们的双曲性。******在我们对拟艾博映射的研究中，我们得到了一般有限情况下的约束全纯曲线，并正在研究代数情况。******我们得到了hyperkähler流形的伪度量的消失，它具有平凡的K_X，并且接近无穷小伪度量