When are braid groups of manifolds Kähler?

Orbifold K{ä}hler Groups related to arithmetic complex hyperbolic lattices

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an {é}tale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.

关于Calabi-Yau流形的超越无基点猜想的一个注记

A note on the transcendental basepoint-free conjecture for Calabi-Yau manifolds

本文证明Calabi-Yau流形的超越无基点猜想在其Beauville-Bogomolov分解的超Kähler因子上成立时即成立，并利用Bakker-Lehn收缩定理在超Kähler流形上给出该猜想成立的一个充分条件。

b_+ = 1的4-流形上的辛一般复结构

Symplectic generic complex structures on 4-manifolds with b_+ = 1

研究了p_g = 0时Kähler曲面上的辛结构.我们给出了一个例子，一个射影曲面，它允许一个辛结构，这是不相容的任何Kähler度量

Some examples due to H. Hironaka

The aim of this paper is to explain the construction by H. Hironaka [H.61] of a holomorphic (in fact "algebraic") family of compact complex manifolds parametrized by $\C$ such for all $s \in \C\setminus \{0\}$ the fiber is projective, but such that the fiber at the origin in non k{ä}hlerian, to mathematicians which are not algebraic geometers. We also explain why it is not possible to make in the same way such an example with fiber at $0$ a simpler example of non k{ä}hlerian Moishezon manifold which is also due to H. Hironaka (see section 5).

Albanese map of special manifolds: A correction

We show that any fibration of a 'special' compact K{ä}hler manifold X onto an Abelian variety has no multiple fibre in codimension one. This statement strengthens and extends previous results of Kawamata and Viehweg when $κ$(X) = 0. This also corrects the proof given in [2], 5.3 which was incomplete.

Very symmetric hyper-Kähler fourfolds

G. Höhn and G. Mason classified all finite groups acting faithfully and symplectically on a hyper-K{ä}hler fourfolds of type K3$^{[2]}$. There are 15 maximal among them, call them $\widetilde{G}_1,\ldots, \widetilde{G}_{15}$. Every manifold of type K3$^{[2]}$ admitting an action of $\widetilde{G}_i$ for some $i$ must necessarily have Picard rank 21 which is maximal. This fact allows us to use lattice-theoretic methods to classify all the finite groups $G$ acting faithfully on a hyper-K{ä}hler fourfold of type K3$^{[2]}$ $X$ such that $G$ contains $\widetilde{G}_i$ as a proper subgroup and $\wi…

On the fundamental groups of effective compact Kähler orbifolds with nef anticanonical bundle

We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]

A criterion for quadraticity of a representation of the fundamental group of an algebraic variety

Let $Γ$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $ρ:Γ\rightarrow G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(Γ, G)$. It is known from the work of Goldman and Millson that if $Γ$ is the fundamental group of a compact K{ä}hler manifold and $ρ$ has image contained in a compact subgroup then $ρ$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(Γ, G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion hol…

凝聚层的导出范畴上的辫群作用

Braid group actions on derived categories of coherent sheaves

本文给出了簇$X$上凝聚层导出范畴上辫群作用的构造。其动机是Kontsevich的同调镜像猜想，以及在辛几何中某些辫群作用的发生。主要结果之一是当$\dim X \geq 2$时，我们的辫群动作总是忠实的。我们描述了光滑和决议的奇异性，导致我们找到的例子所产生的各种奇异性的crepant决议的辫群行动之间的几何镜像对称性。给出了Fano流形上的McKay对应和例外层的关系。此外，还对椭圆曲线的情形作了较详细的计算。

增强规范对称性与辫群作用量

Enhanced gauge symmetry and braid group actions

加强规范对称性出现在II型弦理论（以及F-和M-理论）紧化的卡拉比-丘流形包含例外的因子会议在Dynkin配置。它表明，在许多这样的情况下，在增强的对称点的模量的辫群的行为上的导出范畴的各种层。这个辫群覆盖了增强对称代数的外尔群，它本身以兼容的方式作用于变种的变形空间。扩展这一结果给出了非平凡的$B$ -字段的K3表面上，解释物理限制的$B$ -字段，以及椭圆纤维。本文的观点也为局部$A_{2n}$组态在具有整体$\Z/2$单值性的三重组态中的规范对称性增强提供了新的证据。

Manin conjecture for statistical pre-Frobenius manifolds, hypercube relations and motivic Galois group in coding

This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of Eilenberg--Moore algebras over a Giry monad, the collection $Cap_n$ of all probability distributions on the measurable space $(Ω_n, \mathcal{A})$ (where $Ω$ is discrete with $n$ issues) and it turns out that there exists an embedding relation of Segre type among the product of $Cap_n$'s. We unravel hidden symmetries of these type of embeddings and show that there exists a hypercubic relation. Secondly, we show that t…

幺模双线性格、自同构群、消没循环、单值群、特殊基、辫群作用与来自上三角矩阵的模空间

Unimodular bilinear lattices, automorphism groups, vanishing cycles, monodromy groups, distinguished bases, braid group actions and moduli spaces from upper triangular matrices

本专著从对角线为1的整数上三角矩阵出发，系统研究了秩2和3情形下导出的幺模双线性格、单值群、消没循环、辫群轨道及复流形等结构，并探讨了孤立超曲面奇点情形。