scholarly journals Triviality Properties of Principal Bundles on Singular Curves. II

2020 ◽  
Vol 63 (2) ◽  
pp. 423-433
Author(s):  
P. Belkale ◽  
N. Fakhruddin
Keyword(s):  
Simple Group ◽  
Smooth Curve ◽  
Group Scheme ◽  
Singular Locus ◽  
Cartier Divisor ◽  
Base Change ◽  
Sufficient Condition ◽  
Flat Base ◽  
Simply Connected ◽  
Relative Curve

AbstractFor $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\rightarrow S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D\subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected, but it can be made to vanish by a faithfully flat base change, if $S$ is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

scholarly journals Uniformizing the Moduli Stacks of Global G-Shtukas

2019 ◽  
Author(s):  
E Arasteh Rad ◽  
Urs Hartl
Keyword(s):  
Finite Type ◽  
Moduli Spaces ◽  
Function Field ◽  
Group Scheme ◽  
Base Change ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Moduli Stacks ◽  
Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

scholarly journals On the Boundary Behavior of Conformal Maps

1967 ◽  
Vol 30 ◽  
pp. 83-101 ◽  
Author(s):  
S.E. Warschawski
Keyword(s):  
Boundary Behavior ◽  
Mapping Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Conformal Maps ◽  
Simply Connected Domain ◽  
Definitive Solution ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Fundamental Paper

Suppose Ω is a simply connected domain which is mapped conformally onto a disk. A much studied problem is the behavior of the mapping function at an accessible boundary point P of Ω, in particular the question, under what conditions the map is ‘ “conformai” at such a point (a) in the sense that angles are preserved as P is approached from Ω (“semi-conformality” at P) and (b) the dilatation at P is finite and positive. In his fundamental paper [8] in 1936, A. Ostrowski established a necessary and sufficient condition (depending on the geometry of the domain only) for the validity of the first property which subsumes all previous results and establishes a definitive solution of this problem.

scholarly journals Relation between two twisted inverse image pseudofunctors in duality theory

2014 ◽  
Vol 151 (4) ◽  
pp. 735-764 ◽  
Author(s):  
Srikanth B. Iyengar ◽  
Joseph Lipman ◽  
Amnon Neeman
Keyword(s):  
Finite Type ◽  
Duality Theory ◽  
Inverse Image ◽  
Base Change ◽  
Hochschild Homology ◽  
Fundamental Class ◽  
Compactly Supported ◽  
Canonical Map ◽  
Flat Base ◽  
Derived Functors

Grothendieck duality theory assigns to essentially finite-type maps $f$ of noetherian schemes a pseudofunctor $f^{\times }$ right-adjoint to $\mathsf{R}f_{\ast }$, and a pseudofunctor $f^{!}$ agreeing with $f^{\times }$ when $f$ is proper, but equal to the usual inverse image $f^{\ast }$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.

scholarly journals On base change of the fundamental group scheme

2013 ◽  
Vol 277 (1-2) ◽  
pp. 305-316
Author(s):  
Axel Stäbler
Keyword(s):  
Fundamental Group ◽  
Group Scheme ◽  
Base Change

scholarly journals THE UNIVERSAL COVER OF AN AFFINE THREE-MANIFOLD WITH HOLONOMY OF SHRINKABLE DIMENSION ≤ TWO

2000 ◽  
Vol 11 (03) ◽  
pp. 305-365 ◽  
Author(s):  
SUHYOUNG CHOI
Keyword(s):  
Affine Connection ◽  
Holonomy Group ◽  
Singular Locus ◽  
Universal Cover ◽  
Affine Structure ◽  
Affine Manifold ◽  
Cone Type ◽  
Parallel Volume ◽  
A Cell ◽  
Simply Connected

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to R3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to R3, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to d is d-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to R3, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to R3. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.

scholarly journals FUJITA DECOMPOSITION AND MASSEY PRODUCT FOR FIBERED VARIETIES

2021 ◽  
pp. 1-29
Author(s):  
LUCA RIZZI ◽  
FRANCESCO ZUCCONI
Keyword(s):  
General Type ◽  
Structure Theorem ◽  
Smooth Curve ◽  
Local System ◽  
Base Change ◽  
Smooth Variety ◽  
Massey Product

Abstract Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

An Approach to Higher Teichmüller Spaces for General Groups

2017 ◽  
Vol 2019 (16) ◽  
pp. 4899-4949 ◽  
Author(s):  
Ian Le
Keyword(s):  
Simple Group ◽  
Moduli Spaces ◽  
Cluster Structure ◽  
Type A ◽  
Classical Groups ◽  
Reductive Groups ◽  
Local Systems ◽  
Teichmuller Spaces ◽  
Cluster Varieties ◽  
Simply Connected

Abstract Let $S$ be a surface, $G$ a simply-connected semi-simple group, and $G'$ the associated adjoint form of the group. In Fock and Goncharov [4], the authors show that the moduli spaces of framed local systems $\mathcal{X}_{G',S}$ and $\mathcal{A}_{G,S}$ have the structure of cluster varieties when $G$ had type $A$. This was extended to classical groups in Le [12]. In this article, we give a method for constructing the cluster structure for general reductive groups $G$. The method depends on being able to carry out some explicit computations, and depends on some mild hypotheses, which we state, and which we believe hold in general. These hypotheses hold when $G$ has type $G_2,$ and therefore we are able to construct the cluster structure in this case. We also illustrate our approach by rederiving the cluster structure for $G$ of type $A$. Our goals are to give some heuristics for the approach taken in Le [12], point out the difficulties that arise for more general groups, and to record some useful calculations. Forthcoming work by Goncharov and Shen gives a different approach to constructing the cluster structure on $\mathcal{X}_{G',S}$ and $\mathcal{A}_{G,S}$. We hope that some of the ideas here complement their more comprehensive work.

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