Complex geometry of orbifold pairs and of their moduli spaces; structure, classification and relation to arithmetic geometry

我们从双曲性的角度研究代数簇。我们使用各种方法来约束曲线和获得Kobayashi伪距的正性(即双曲性)，并使用现代代数几何来获得它的消失，其中丰度猜想是中心焦点。最近，我们也开始研究簇的模空间，特别是具有固定嵌入到射影空间的流形。我们已经开始了复代数几何中这一观点的复兴，并将通过组织活动和培养HQP来传播这条有希望的道路。我们最近的焦点集中在典型类K是NEF的簇，包括没有有理曲线的簇和具有半负全纯曲率的簇(根据我们的结果)。在我们过去的建议中，我们已经得到了KLT型奇异簇的Bogomolov-Miyaoka-Yau不等式以及它们在相等情况下的一致化，我们的目标是更奇异的Lc情形，以作为丰度问题的引线。证明了半负全纯对分曲率的射影Kähler流形被一个交换簇与一个充足K簇的乘积所覆盖。我们的目的也是为了同样的目的，去掉“二等分”，这将验证这种情况下的丰度猜想，更一般地，通过我们关于几乎阿贝尔原纤维的结果，对于没有有理曲线的光滑变种。一个期望的成分是这样一个具有平凡K的簇被一个交换簇覆盖，我们在半负全纯曲率和一般的Aim的情况下证明了这一点。S·S·Kobayashi猜想双曲簇有充足的K。没有有理曲线的射影簇的类比本质上是Mori弯曲断裂定理。我们解决了对数DLT对的拟投射情形下的类比猜想，给出了这一广义情形下Mori锥定理的几何版本。在这种情况下，我们还解决了小林的猜想，模上所希望的成分和丰度猜想，这两个猜想都知道直到三维。我们正在开发处理这些奇异变量的新方法，以获得关于线性系统的精确结果。Kobayashi在Kähler世界的猜想已经被S.T.Yau等人解决了。部分使用我们的技术。它说，负全纯曲率的射影Kähler流形有充足的K。在非Kähler世界中，曲面结果是已知的模一类VII0曲面，我们正在与专家Apostolov和Dloussky研究这类曲面。在我们对拟阿尔巴尼亚映射的研究中，我们得到了一般有限情形下的约束全纯曲线，并且正在计算代数情形。我们得到了超Kähler流形的伪距的消失，并且正在快速地逼近无穷小伪距。

We study algebraic varieties from the hyperbolicity perspective. We use various methods including Nevanlinna theory, Ahlfors-Schwarz lemmas, jets, etc, for constraining curves and for getting positivity of the Kobayashi pseudometric (i.e. hyperbolicity) and use modern algebraic geometry for getting its vanishing, the abundance conjecture being a central focus. More recently, we started also to look at the same for moduli spaces of varieties, specifically of manifolds with a fixed embedding into projective space. We have started a revival in this perpective in complex algebraic geometry and will propagate this promising path by organizing activities on them and by fostering of HQPs. Our recent focus centres on varieties whose canonical class K are nef, including varieties without rational curves and (by our results) varieties with seminegative holomorphic curvature. Having obtained the Bogomolov-Miyaoka-Yau inequality for singular varieties of klt type and their consequent uniformization in the case of equality, a project in our past proposal, we aim for the more singular lc case for a lead on the abundance problem. 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 an ample K variety. We aim for the same by dropping "bisectional", which would verify the abundance conjecture in this case, and more generally for smooth varieties without rational curves via our results on almost abelian fibrations. A hoped-for ingredient is that such a variety with trivial K be covered by an abelian variety, which we verified in the case of sem-inegative holomorphic curvature and aim in general. S. Kobayashi conjectured that a hyperbolic variety has ample K. The analog for a projective variety without rational curves is in essence Mori bend-and-break theorem. We have resolved the analog conjecture in the quasiprojective setting of log dlt pairs, providing a geometric version of Mori's cone theorem in this generalized setting. We have also resolved in this case Kobayashi's conjecture modulo the above hoped-for ingredient and the abundance conjecture, both known up to dimension three. We are exploiting new methods for these singular varieties for sharp results on linear systems. Kobayashi's conjecture in the Kähler world has been resolved by S.T. Yau et al. partially using our techniques. It says that a projective Kähler manifold of negative holomorphic curvature has ample K. In the non-Kähler world, the surface result is known modulo a class of VII0 surfaces for which we are investigating with the experts Apostolov and Dloussky. 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 hyperkählers, manifolds with trivial K, and are rapidly closing in on the infinitesimal pseudometric.