Generalized hyperbolicity and the geometry of algebraic varieties

我们从双曲性的角度研究代数簇。我们利用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Kobayashi伪距(即反双曲性)的正性(分别为零)。在这项研究中，我们还使用了代数几何中的现代工具，其中森喜朗的丰度猜想是一个中心焦点。我们已经在复杂几何和代数几何中开始了这方面的复兴，并旨在通过进一步的有组织的活动和培养HQP来继续这条有希望的道路。 我们最近的焦点集中在标准类K_X是NEF的簇X，包括那些没有有理曲线的簇和那些全纯截曲率H_X是半负的簇。在得到了矩阵矩阵中一类自然奇异变元的Bogomolov-Miyaoka-Yau不等式以及它们在相等情况下的一致化后，我们的目标是为丰度猜想寻找最奇异的这类变元。 G.Liu在郑福生工作的基础上证明了具有半负全纯对分曲率的射影Kähler流形被具有充足K_X的交换簇的乘积所覆盖。我们的目标是对半负H_X的情形也是如此，更一般地，对于没有有理曲线的光滑簇X，通过我们关于几乎交换函数的结果，这将证实这两种情况下的充分性。一个期望的成分是这样一个具有平凡K_X的簇X被一个阿贝尔簇覆盖，我们在半负H_X和AIM的情况下证明了这一点。 S.Kobayashi猜想双曲簇X有充足的Kx。没有有理曲线的射影簇的模拟是Mori折断定理。我们解决了DLT对的最优奇异设置下的类比猜想，在这种更一般的设置下提供了Mori锥定理的几何版本。在此背景下，我们还解决了小林猜想模上所希望的成分和丰度猜想，这两个猜想都是已知的，直到三维。我们正在开发我们的新方法，以获得关于线性系统的一般精确结果。 Kobayashi在Kähler世界的猜想已经被S.T.Yau等人解决了。部分使用了我们的技术。它说一个带有H_X的射影Kähler X

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