Generalized hyperbolicity and the geometry of algebraic varieties

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。

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