Oliver Bültel
We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a consequence (of our theorem 9.1) one obtains that for every prescribed odd prime characteristic $p$ every bounded symmetric domain possesses quotients by arithmetic groups whose models have good reduction at a prime divisor of $p$.
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We construct the fine moduli space of log abelian varieties with PEL structure, which gives a toroidal compactification of the moduli space of abelian varieties with PEL structure.
Serre-Tate Theory for Moduli Spaces of PEL Type
本文的主要目的是将Serre-Tate的“普通”局部模理论推广到PEL型的Shimura簇。为此，我们开发了一个广义的概念，平凡性，我们证明了一些基本结果，我们研究了普通物体的形式变形。一般来说，形式变形空间得到一个新的“类群”结构，我们称之为“级联”。进一步给出了关于PEL模空间的Ekedahl-Oort分层的一些结果，以及我们的结果在同余关系研究中的应用.
We study $p$-adic integral models of certain PEL Shimura varieties with level subgroup at $p$ related to the $Γ_1(p)$-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of $\mathbb{Q}_p$ and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are étale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort-Tate generator of a group scheme. We use these local models t…
Character Varieties and the Moduli of Quiver Representations
设G是一个李群，Q是一个有关系的群.本文定义了Q的G-值表示，它直接推广了有限生成群的G-值表示。虽然作为G-空间，G-值商表示比有限生成群的G-值表示更一般，但我们通过折叠箭头证明了它们的商空间是等价的。然后，我们建立了一个一般的标准，G-值的表示的模的关系，允许一个强形变收缩的紧商，通过捏顶点上。这提供了两个不同的推广的主要结果在我们以前的工作。最后，我们建立了GL（n，C）和SL（n，C）值的正则表示的模空间嵌入到传统的具有常维数向量的正则表示的模空间中的正则理论条件.
Infinitesimal automorphisms and obstruction theory on the moduli of $L$-valued $G$-Higgs bundles
研究 $L$-valued $G$-Higgs 包的微分自同构及稳定性模堆结构，构造对称完美障碍理论以定义 Vafa-Witten 不变量。
Hurwitz Spaces and Moduli Spaces as Ball Quotients via Pull-back
利用局部系统中的交同调定义了超几何函数。拓扑结构被强调;分析只通过霍奇分解进入一次。通过拉回过程，我们构造了Deligne-Mostow模空间DM（n，mu）的特殊子集S_{pi}，它是由Hurwitz空间导出的.已知某些DM（n，μ）是球幂次，由在复球中取值的超几何函数均匀化（即，复双曲空间）。给出了S_{pi}是子球商的充分条件。详细分析最简单的例子，我们描述了球商结构附加到一些非齐次二元形式的模空间。这恢复特别是结构上的模空间的合理椭圆曲面给出的Heckman和Looijenga。我们利用自然的偏序的Deligne-Mostow的例子（这给了一个简单的方法，看到原来的列表Mostow，最终纠正瑟斯顿，是错误的），因此突出两个关键的例子，我们称之为高斯和爱森斯坦祖先的例子。
We use the gradients of theta functions at odd two-torsion points --- thought of as vector-valued modular forms --- to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of two-torsion points.
A dimension formula for Ekedahl-Oort strata
研究了PEL型模空间上的Ekedahl-Oort分层。这些层用外尔群中的陪集来索引。本文的主要结果是：地层的维数等于相应的外尔群元的长度。我们还讨论了一些明确的例子。
Lagrangian structures on the derived moduli of constructible sheaves
本文研究了紧致定向流形上 $\mathcal{D}(k)$-valued 构造sheaves 和 perverse sheaves 的模形，证明其具有 $(2-n)$-移位的拉格朗日结构，并通过立方体的松散粘合结果实现。
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism …
$\mathbb{A}^1$-connectedness of moduli of semistable bundles and symplectic bundles on a curve
本文证明无限域上光滑射影曲线半稳定向量丛（固定行列式）及辛向量丛模栈的 $\mathbb{A}^1$-连通性，条件依赖于有理点存在，并推广至拟抛物情形。
Let $\mathcal{M}(d,χ)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $χ$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We compute the $A$-valued motivic measure $μ_A(\mathcal{M}(d,χ))$ of $\mathcal{M}(d,χ)$ and get explicit formula in codimension $D:=ρ_d-1$, where $ρ_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,χ)$ when $d$ is coprime to $χ$.
Topological flatness of local models for ramified unitary groups. I. The odd dimensional case
局部模型是根据线性代数模问题定义的某些方案，其给出由Rapoport和Zink定义的某些p-adic PEL Shimura簇的积分模型的étale局部邻域。当定义志村簇的群在p处分歧时，最初定义的局部模型（以及志村模型）可能不是平坦的，并且需要修改它们的定义以获得平坦的方案。在酉相似群在Q_p处的局部化是分歧的情况下，准分裂GU_n，Pappas和Rapoport在定义原局部模型的模问题上增加了新的条件，即所谓的楔条件和旋条件，并证明了新的局部模型是平坦的。我们证明了他们猜想的一个初步形式，即他们的新模型是拓扑平坦的，在n是奇数的情况下。
Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves
该研究针对曲面上的希格斯层模空间与希钦映射，证明了高维谱簇的Noether–Lefschetz型定理，给出一般希钦纤维非空的充要条件，并分析第二陈类变化对模空间几何的影响。
For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\bar{k}$-valued points of moduli spaces of quiver representations over $k$; the fixed locus is the set of $k$-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of $k$. We provide a modular interpretation of this decomposition using quiver representations over division algebras, and we reinterpret this description using twisted quiver representations. We also see that moduli spaces of twisted quiver representations give different forms of the moduli space…
Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP^1 into G_r(C^n)
本文给出了Grassmannian G_r（C^n）上以及从球面到Grassmannian G_r（C^n）的固定度d的全纯映射的模空间上的算子的显式表达式。在Grassmannain上得到的相关器是Schubert自相交公式的第一步推广。用两个封闭的公式给出了r= 2，3时模空间上的交数，当r=2时，交数产生交错Fibonacci数、Pell数和一般的粒子在有吸收障碍的直线上的随机行走.当r=3时，交点数形成了一个组织良好的模式。
Let $K$ be a complete algebraically closed non-archimedean valued field of characteristic zero, and let $X$ be a finite type scheme over $K$. We say $X$ is $K$-analytically Borel hyperbolic if, for every finite type reduced scheme $S$ over $K$, every rigid analytic morphism from the rigid analytification $S^{\mathrm{an}}$ of $S$ to the rigid analytification $X^{\mathrm{an}}$ of $X$ is algebraic. Using the Viehweg-Zuo construction and the $K$-analytic big Picard theorem of Cherry-Ru, we show that, for $N \geq 3$ and $g \geq 2$, the fine moduli space $\mathcal{M}^{[N]}_{g,K}$ over $K$ of genus $…
Volumes of moduli spaces of hyperbolic surfaces with cone points
该研究计算了带测地、尖点及锥边界的双曲曲面模空间在大锥角等新情形下的体积，为代入虚边界长度的Mirzakhani多项式赋予几何意义，并分析了锥角趋近2π时的体积行为。
Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two
本文结合局部系统上同调描述与 Euler 特征计算，求得了二阶二级向量值 Siegel 模形式空间在 6 个字母对称群下的同型分解。
Universality theorems for configuration spaces of planar linkages
利用平面连杆机构的模空间证明了向量值多项式映射、实代数集和紧光滑流形的可实现性定理。我们还建立了机械连杆机构的模空间的普适性定理与射影配置之间的关系。
Multiplicative Hitchin systems are analogues of Hitchin's integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is group-valued, rather than Lie algebra valued. We discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where C = CP1 with a fixed framing at infinity. In this case we prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperkähler …
The classical theory of the cross-ratio is a beautiful case study of the moduli of ordered points of the projective line and of invariants of the action of $PGL_2$. We generalize the theory of the cross-ratio to the setting of $S$-valued points for an arbitrary scheme $S$. To accomplish this goal, we provide a comprehensive and computationally focused treatment of automorphisms of projective space over $S$, of equalizers in the category of schemes, and of vanishing loci of sections of line bundles. Most of these ideas exist in the literature, though not with the level of detail or generality t…
Higher Arf Functions and Topology of the Moduli Space of Higher Spin Riemann Surfaces
证明了紧致Riemann曲面上的m-自旋结构空间的连通分支同胚于向量空间R^d的离散群作用的商.我们的证明是基于黎曼曲面上的m-自旋结构空间表示为曲面的基本群上的Z/mZ-值函数的有限仿射空间。
Connected components of the moduli space of L-parameters
本文证明了关于p-adic群的L-参数模空间连通分支与R-表示范畴中块相关的一个强形式猜想，适用于任何Zbar[1/p]上的整环R。
$p$-adic hyperbolicity for moduli spaces of abelian motives
本研究证明阿贝尔型Shimura簇在离散赋值域上满足$p$进Borel延拓性质，推导了其在多领域的应用，并证明了Rapoport-Zink空间的延拓结果。
Differential forms on universal K3 surfaces
该研究针对带点K3曲面模空间光滑射影模型的全纯微分形式，证明了特定k值下形式消失，并给出其余情形与向量值/标量值模形式的同构关系，结果适用于格极化的一般情形。
We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.
Single-valued periods of meromorphic modular forms and a motivic interpretation of the Gross-Zagier conjecture
本文通过动机理论为 Gross-Zagier 猜想提供几何解释，证明其可由动机理论中的标准猜想推出，并构造矩阵值高阶 Green 函数作为椭圆曲线模叠动机的单值周期，从而提出该猜想的广泛推广。
We discuss the Kähler quantization of moduli spaces of vortices in line bundles over compact surfaces $Σ$. This furnishes a semiclassical framework for the study of quantum vortex dynamics in the Schrödinger-Chern-Simons model. We follow Deligne's approach to Quillen's metric in determinants of cohomology to construct all the quantum Hilbert spaces in this context. An alternative description of the quantum wavesections, in terms of multiparticle states of spinors on $Σ$ itself (valued in a prequantization of a multiple of its area form), is also obtained. This viewpoint sheds light on the natu…
A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2,C)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance b…
Finiteness of function field-valued points on exceptional Shimura varieties
研究证明了例外Shimura品种上函数域值点的有限性，并推广到更一般的Shimura品种和素数条件。
For the TZ metric on the moduli space $\mathscr{M}_{0,n}$ of $n$-pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space $\mathfrak{S}_{g,n}$ as holomorphic fibration $\mathfrak{S}_{g,n}\rightarrow\mathfrak{S}_{g}$ over the Schottky space $\mathfrak{S}_{g}$ of compact Riemann surfaces of genus $g$, where the fibers are configuration spaces of $n$ points. For the tautological line bundles $\mathscr{L}_{i}$ over $\mathfrak{S}_{g,n}$ we define Hermitian metrics $h_{i}$ in terms of Fourier coefficients of a coveri…
Instantons on Gravitons
ALE引力瞬子上的Yang-Mills瞬子是由Kronheimer和Nakajima用满足代数方程的矩阵构造的。这些都被方便地组织成一个小册子。我们在ALF引力瞬子上构造了一般的杨-米尔斯瞬子。我们的数据是用单变量的矩阵值函数来表示的，这些函数又被组织成一个弓形。我们介绍了弓的一般概念，它的表示，它的相关数据和模空间的解决方案。对于一个明智选择的弓，Nahm变换将任何弓的解映射到ALF空间上的瞬子。我们证明了这个映射尊重模空间上的所有复结构，因此它很可能是等距的，并利用这一事实研究了ALF空间上瞬子的模空间的渐近性。
Let ${\mathcal B}_g(r)$ be the moduli space of triples of the form $(X,\, K^{1/2}_X,\, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^*{\mathcal B}_g(r)$--torsor ${\mathcal H}_g(r)$ over ${\mathcal B}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundle…
Graph enumeration for moduli spaces of curves and maps
本文提出基于图枚举的微积分，用于计算图形分层模空间的等变动机不变量，并应用于Deligne--Mumford模空间及稳定映射空间，借助Pólya--Petersen特征和花环积对称函数代数推导新公式。
Singularities of free group character varieties
设X是秩为r的自由群的SL（n，C），SU（n），GL（n，C）或U（n）-值表示的模空间.我们对X的代数奇异分层进行了分类。这归结为表明，如果存在奇点，奇异轨迹正好对应于可约表示。然后通过将代数奇点与拓扑奇点联系起来，我们证明了模空间X一般不是拓扑流形，除了我们明确描述的几个例子。
Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $λ$-connection …
In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs…
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