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

我们从双曲的角度研究代数变分。我们使用各种方法，包括Nevanlinna理论，Ahlfors-Schwarz引理，射流等，来约束曲线并获得Kobayashi伪度量的正性（即双曲性），并使用现代代数几何来获得其消失，丰度猜想是中心焦点。最近，我们也开始研究变种的模空间，特别是在射影空间中固定嵌入的流形。我们已经开始在复杂代数几何中复兴这一观点，并将通过组织相关活动和培养hqp来传播这一有前途的途径。我们最近的焦点集中在正则类K为nef的变种，包括没有有理曲线的变种和（根据我们的结果）具有半负全纯曲率的变种。我们已经获得了klt型的奇异变种的Bogomolov-Miyaoka-Yau不等式，以及它们在相等情况下的一致化，我们的目标是在丰度问题上取得更奇异的lc情况。G. Liu在郑F.的基础上证明了半负全纯对分曲率的射影Kähler流形被一个阿贝尔变体与一个充足的K变体的乘积所覆盖。我们的目标是通过放弃“对分”，这将在这种情况下验证丰度猜想，并且通过我们在几乎阿贝尔颤振上的结果更普遍地验证没有有理曲线的光滑品种。一个期望的成分是这样一个具有平凡K的变体被一个阿贝尔变体所覆盖，我们在半负全纯曲率的情况下验证了这一点，并以一般的目标为目标。S. Kobayashi推测双曲型有足够的k值。无有理曲线的射影型的类比本质上是Mori弯曲-断裂定理。我们解决了log dlt对拟射影情况下的类似猜想，在这种广义情况下给出了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。我们正在开发新的方法来处理这些奇异变量，以便在线性系统上得到尖锐的结果。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。它说负全纯曲率的投影Kähler流形有足够的k。在non-Kähler世界中，曲面结果是已知模于一类VII0曲面的，我们正在与专家Apostolov和Dloussky一起研究。在拟艾博映射的研究中，我们给出了一般有限情况下的约束全纯曲线，并给出了代数情况下的约束全纯曲线。我们得到了具有平凡K的hyperkählers流形的伪度量的消失性，并迅速逼近无穷小伪度量。

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.