Hyperbolicity of geometric orbifolds
Erwan Rousseau
本文在F.坎帕纳推广经典方法，在这种情况下，我们得到退化声明的情况下，没有第二大定理是已知的值分布理论的整个曲线与分歧。
We study complex hyperbolicity in the setting of geometric orbifolds introduced by F. Campana. Generalizing classical methods to this context, we obtain degeneracy statements for entire curves with ramification in situations where no Second Main Theorem is known from value distribution theory.
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Hyperbolicity of smooth logarithmic and orbifold pairs in $\mathbb{P}^n$
研究$\mathbb{P}^n$中超平面排列的条件，以确定关联的对数余切丛在边界上是 ample 的，并扩展到轨道设置，应用超曲面性结果，改进Darondeau-Rousseau的结论。
We define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green-Griffiths-Lang conjecture. This contrasts with an important result of Demailly (2010) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establi…
We prove that the Green--Griffiths--Demailly (GGD) hyperbolicity thresholds are structurally invariant. In other words, the minimal jet order and asymptotic growth rate at which invariant jet differentials appear remain unchanged when passing from a compact complex manifold to any compact smooth analytic Deligne--Mumford stack (orbifold) with the same coarse Kähler class. We establish an orbifold Riemann--Roch formula showing that only the identity sector contributes to the leading $m^n$ term of the Euler characteristic $χ$, while all twisted sectors contribute only $O(m^{n-1})$. Together with…
Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
本文证明光滑复三次曲面模空间的轨道基本群是层级双曲群，利用不完备复双曲度量，并推广至其他模空间。
Kobayashi hyperbolicity of complete linear systems on abelian varieties
研究阿贝尔簇上完全线性系统中非常一般元素的柯拜ashi超曲率条件，并提出相关猜想。
Non-reductive geometric invariant theory and hyperbolicity
本文利用非约化几何不变量理论商的相交理论及 Riedl 和 Yang 的工作，证明了一般超曲面的 Green--Griffiths--Lang 和 Kobayashi 双曲性猜想。
In this paper, we introduce a sub-family of the usual generalized Wronskians, that we call geometric generalized Wronskians. It is well-known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global f…
Orbifoldes speciales et classification bimeromorphe des varietes kaehleriennes compactes
这是[Ca 01]=math.AG/0110051的续篇。我们定义了几何orbifolds的双纯{ \it范畴}。这些插值（紧K\”ahler）流形和这样的流形与对数结构。这些几何orbifolds被认为是从他们的几何观点，从而配备了通常的不变量品种：态射和双纯映射，微分形式，基本群和普遍覆盖，领域的定义和有理点。最基本的性质，直接适用于没有orbifold结构的品种的情况下，在这里建立。[Ca 01]的参数可以直接适用于将主要结构结果扩展到这个轨道范畴。我们希望以后再回到更深的方面。其动机是紧致K\”ahler分类理论的自然框架（和复射影）流形至少包括orbifolds的范畴，如[Ca 01]中所示，通过{ \it special}流形作为具有$κ_+=-\infty$或$κ=0$的orbifolds的塔的函数分解，并且似乎还通过极小模型程序，其中大多数证明工作后，只有一个“边界”的附加。此外，纤维化在几何轨道折叠的双半纯范畴中享有在没有轨道折叠结构的变种范畴中不满足的扩展性质，允许根据一般纤维和基底的不变量来表达全空间的不变量。例如，基本群的自然序列在那里是正合的;如果类属纤维和基是特殊的，那么全空间也是特殊的。这使得该类别适合于将具有$κ_+=-\infty$或$κ=0$的轨道的性质提升到特殊的轨道。
Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion f…
Special orbifolds and birational classification: a survey
我们将展示如何通过函子和正则纤维化分解任意$n$维复射影{虽然几何结果适用于紧致K\”ahler流形而没有变化，但为了简单起见，我们在这里只考虑这种特殊情况。}将$X$变种为变种（或者更确切地说，是由规范丛的“符号”（负、零或正）确定的三种\pure几何之一的“几何orbifolds”\rq $)$。这些分解是双有理不变的，将考虑基于\canonical（或“Kodaira \rq $)$维）的这些\pure几何的双有理版本。这些分解的一个关键特征确实是，为了处理纤维化的多个纤维，它们需要在更大的“几何轨道”$(X| Δ)$类别中发生。这些是各种“虚拟分支覆盖物”，"实际上消除“了纤维化的多个纤维。虽然形式上与LMMP的“对”相同（参见\cite{kmm}、\cite{KM}、\cite{BCHM}和那里的参考文献），但它们在这里是完全几何的对象，配备了通常的几何不变量的变种，如（对称）微分形式层、基本群、小林伪度量、整点、态射和有理映射。本文旨在尽可能简明扼要地揭示（加上一些附加的主题或发展）分别发表在ANN. Inst. Fourier（2004）和J.Inst. Math. Jussieu上的arXiv：math/0110051和arXiv：0705.0737的主要内容，基本上跳过了证明。
Geometric realizability of epimorphisms to curve orbifold groups
该研究在曲线轨形群的轨形欧拉示性数为负时，证明了紧Kähler流形Zariski开集的基本群商到该群的满同态可由满全纯映射几何实现，并将结果应用于Serre相关问题。
On orbifolds and free fermion constructions
这项工作发展了轨道折叠和自由费米子模型之间的对应关系。本文给出了轨道折叠X/G的一个完全分类，X是三条椭圆曲线的乘积，G是作用在X上的扭曲群（Z_2）^2的阿贝尔扩张。每个这样的商X/G示出给一个适当的自由费米子模型，包括几何NAHE+模型的几何解释。然而，半现实的NAHE自由费米子模型被证明是非几何的：它的霍奇数不能被任何轨道X/G再现。在特殊情况下，它表明，X/G可以同意一些Borcea-Voisin三倍，一个orbifold限制的Schoen三倍，和几个进一步的orbifolds。这产生了自由费米子模型与几何解释这样特殊的三倍。
We consider the question of how geometric structures of a Deligne-Mumford stack affect its Gromov-Witten invariants. The two geometric structures studied here are {\em gerbes} and {\em root constructions}. In both cases, we explain conjectures on Gromov-Witten theory for these stacks and survey some recent progress on these conjectures.
Extension of adapted differentials on klt orbifolds
针对 Campana 意义下的几何轨形，证明当轨形为 klt 时，其适当分歧覆盖上的适配自反微分可延拓为覆盖奇点解消上的正则微分形式，解决适配微分的延拓问题。
Proven by A. Parshin and S. Arakelov in the early 70's, Shafaverich hyperbolicity conjecture states that a family of curves of genus $g\ge2$ parametrized by a non hyperbolic curve (\emph{i.e.} isomorphic to $\mathbb{P}^1$, $\mathbb{C}$, $\mathbb{C}^*$ or an elliptic curve) has to be isotrivial : the moduli of smooth fibres are constant. In higher dimensions, Viehweg's works on moduli of canonically polarized manifolds led him to generalize this statement in the following way: if a family of canonically polarized manifolds (parametrized by a quasi-projective base) has maximal variation, the bas…
The map to the orbifold base need not be an orbifold map
本文通过一个例子展示了到 orbifold 基的映射不必为 orbifold 映射，并探讨了其对 Campana 猜想的影响。
Gerbes over Orbifolds and Twisted K-theory
本文构造了轨道$X$上格贝群的显式几何模型。我们展示了如何从它的曲率，我们可以得到它的示性类在$H^3(X)$通过Chern-Weil理论。对于任意的几何体$\LL$，构造了$X$的轨道折叠$K$ -理论的扭转$\Korb{\LL}(X)$，推广了Rosenberg，Witten，Atiyah-Segal和Bowknegt等人的扭转. Adem-Ruan等人在光滑情况下和Orbifold上的离散挠率。
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