Algebraic Geometry: Salt Lake City 2015最新文献

英文中文

How often does the Hasse principle hold? 哈塞原则多久成立一次?

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.2/01700

T. Browning

We survey recent efforts to quantify failures of the Hasse principle in families of rationally connected varieties.

我们调查了最近在合理连接品种族中量化Hasse原理失效的努力。

引用次数: 11

Hall algebras and Doanldson-Thomas invariants 霍尔代数与doanlson - thomas不变量

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/pspum/097.1/03

T. Bridgeland

引用次数: 5

Non-commutative deformations and Donaldson-Thomas invariants 非交换变形与Donaldson-Thomas不变量

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.1/01687

Yukinobu Toda

引用次数: 3

Θ-stratifications, Θ-reductive stacks, and applications Θ-stratifications、Θ-reductive堆栈和应用程序

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.1/01678

Daniel Halpern-Leistner

These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the “beyond geometric invariant theory” program for analyzing moduli problems in algebraic geometry. We discuss methods for analyzing stability in general moduli problems, focusing on the moduli of coherent sheaves on a smooth projective scheme as an example. We describe several applications: a general structure theorem for the derived category of coherent sheaves on an algebraic stack; some results on the topology of moduli stacks; and a “virtual non-abelian localization formula” in K-theory. We also propose a generalization of toric geometry to arbitrary compactifications of homogeneous spaces for algebraic groups, and formulate a conjecture on the Hodge theory of algebraic-symplectic stacks. We present an approach to studying moduli problems in algebraic geometry which is meant as a synthesis of several different lines of research in the subject. Among the theories which fit into our framework: 1) geometric invariant theory, which we regard as the “classification” of orbits for the action of a reductive group on a projective-over-affine scheme; 2) the moduli theory of objects in an abelian category, such as the moduli of coherent sheaves on a projective variety and examples coming from Bridgeland stability conditions; 3) the moduli of polarized schemes and the theory of K-stability. Ideally a moduli problem, described by an algebraic stack X, is representable by a quasi-projective scheme. Somewhat less ideally, but more realistically, one might be able to construct a map to a quasi-projective scheme q : X→ X realizing X as the good moduli space [A] of X. Our focus will be on stacks which are far from admitting a good moduli space, or for which the good moduli space map q, if it exists, has very large fibers. The idea is to construct a special kind of stratification of X, called a Θ-stratification, in which the strata themselves have canonical modular interpretations. In practice each of these strata is closer to admitting a good moduli space. Given an algebraic stack X, our program for analyzing X and “classifying” points of X is the following: (1) find a Θ-reductive enlargement X ⊂ X′ of your moduli problem (See Definition 2.3), (2) identify cohomology classes ` ∈ H2(X′;Q) and b ∈ H4(X′;Q) for which the theory of Θ-stability defines a Θ-stratification of X′ (See §1.2), (3) prove nice properties about the stratification, such as the boundedness of each stratum. We spend the first half of this paper (§1 & §2) explaining what these terms mean, beginning with a detailed review of the example of coherent sheaves on a projective scheme. Along the way we discuss constructions and results which may be of independent interest, such as a proposed generalization of toric geometry which replaces fans in a vector space with certain collections of rational polyhedra in the spherical building of a reductive

这是2015年美国科学院暑期代数几何学院一次同名讲座的扩展笔记。介绍和概述了用于分析代数几何中模问题的“超越几何不变理论”程序。讨论了一般模问题的稳定性分析方法，以光滑投影格式上相干轴的模为例。我们描述了几个应用:代数堆栈上相干束的派生范畴的一般结构定理;关于模栈拓扑的一些结果以及k理论中的“虚非阿贝尔局部化公式”。我们还提出了将环几何推广到代数群的齐次空间的任意紧化，并提出了关于代数-辛堆的Hodge理论的一个猜想。我们提出了一种研究代数几何中模问题的方法，这意味着它是该主题中几个不同研究方向的综合。在适合我们的框架的理论中:1)几何不变理论，我们将其视为约化群在投影-上仿射方案上作用的轨道的“分类”;2)阿贝尔范畴中对象的模理论，如射影变化上相干束的模和来自布里奇兰稳定条件的例子;3)极化格式的模和k稳定理论。理想情况下，由代数堆栈X描述的模问题可以用拟投影格式表示。有些不太理想，但更现实的是，人们可能能够构造一个映射到拟射影格式q: X→X，实现X作为X的良模空间[a]。我们的重点将放在远不能允许良模空间的堆栈上，或者对于良模空间映射q，如果存在，具有非常大的纤维。这个想法是构建一种特殊的X层，称为Θ-stratification，其中的地层本身具有规范的模块化解释。在实践中，这些层中的每一个都更接近于允许一个好的模空间。给定一个代数堆X，我们用于分析X和对X的点进行“分类”的程序如下:(1)找到模问题的Θ-reductive放大X´X´(见定义2.3)，(2)识别上同类'∈H2(X ';Q)和b∈H4(X ';Q)，其中Θ-stability理论定义了X '的Θ-stratification(见§1.2)，(3)证明关于分层的良好性质，例如每个层的有界性。我们将在本文的前半部分(§1和§2)解释这些术语的含义，首先详细回顾投影格式上相干束的例子。在此过程中，我们讨论了可能独立感兴趣的构造和结果，例如在约化群G的球形建筑中用某些有理多面体集合取代矢量空间中的扇形的环形几何的建议推广(§2.2)。在本文的第二部分，我们讨论了Θ-stratifications的应用。在(§3和§4)中，我们讨论了如何使用派生范畴来分类柯文上同的满射定理(见定理3.1)，以及这个主题的几种变体。具体地说，我们讨论了如何使用派生代数几何方法和Θ-stratifications理论来建立具有Θ-stratification的派生堆类的结构定理(定理3.17，定理3.22)，并以此证明了Borel-Moore同构的Kirwan满射的一个版本(推论4.1)。作为一个应用，我们证明了(定理4.3)的Borel-Moore同调的庞加莱多项式

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引用次数: 4

Moduli of stable log-varieties–An update 稳定对数的模数-变量-更新

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/pspum/097.1/14

Sandor J. Kovacs

引用次数: 0

Bi-algebraic geometry and the André-Oort conjecture 双代数几何与andr<s:1> - oort猜想

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.2/01709

B. Klingler, E. Ullmo, A. Yafaev

引用次数: 33

Notes on homological projective duality 关于同调射影对偶的注释

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.1/01686

Richard P. Thomas

引用次数: 17

Principal bundles and reciprocity laws in number theory 数论中的主束和互易律

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.2/01708

Minhyong Kim

We give a brief survey of some ideas surrounding non-abelian Poitou-Tate duality in the setting of arithmetic moduli schemes of principal bundles for unipotent fundamental groups and their Diophantine applications. 1. Principal bundles and their moduli Moduli spaces of principal bundles (or torsors) have played a prominent role in geometry, topology, and mathematical physics over the last half-century [2, 12, 21, 25]. However, it would appear that arithmetic applications predate these developments by many decades. A prominent example is Weil’s work on the Jacobian JX of an algebraic curve X [23]. While its analytic construction had been known since the 19th century, Weil gave an algebro-geometric construction so that the inclusion X ⊂ JX that sends x to the class of the line bundle OX(x)⊗OX(−b) might be used to study the arithmetic of X. In Weil’s approach, when X is defined over a number field F , so is JX . Furthermore, choosing an F -rational basepoint b ∈ X(F ), rationality is preserved by the inclusion, suggesting the possibiity of studying X(F ) via the superset JX(F ). This research resulted in the Mordell-Weil theorem, stating that JX(F ) is finitely-generated, a result which then was generalised to arbitrary abelian varieties. Weil hoped to prove that the geometric intersection X ∩ JX(F ) is finite, thereby proving the Mordell conjecture. However, the abelian nature of JX(F ), a useful property in itself, turned out to be an obstruction more than a help when applied to the arithmetic of X. Nevertheless, the Jacobian was subsequently used by Siegel to prove the finiteness of integral points on affine curves over number fields, thereby convincing arithmeticians of the utility of this abstract construction. Later, Weil attempted to move beyond the abelian framework by considering moduli spaces Bunn(X) of vector bundles of rank n over X [24]. Serre [20] describes this work in his obituary for Weil as ‘a text presented as analysis, whose significance is essentially algebraic, but whose motivation is arithmetic.’ He correctly stresses the visionary nature of the paper, written long before the advent of geometric invariant theory made it possible to give a systematic treatment of such moduli spaces. Today, they play an important role in various geometric versions of 1991 Mathematics Subject Classification. 14G10, 11G40, 81T45 . Supported by grant EP/M024830/1 from the EPSRC. c ©0000 (copyright holder)

本文简要讨论了单幂基本群的主束的算术模格式集合中关于非阿贝的Poitou-Tate对偶的一些思想及其丢凡图应用。1. 在过去的半个世纪里，主束及其模束(或旋量)的模空间在几何、拓扑和数学物理中发挥了突出的作用[2,12,21,25]。然而，算术应用似乎比这些发展早了几十年。一个突出的例子是Weil对代数曲线X的雅可比矩阵JX的研究[23]。虽然它的解析结构早在19世纪就为人所知，但Weil给出了一个代数几何结构，使得包含X∧JX将X送到线束OX(X)⊗OX(−b)的类中，可以用来研究X的算术。在Weil的方法中，当X定义在一个数字域F上时，JX也定义在F上。进一步，选取一个F -有理基点b∈X(F)，其包含保留了合理性，提示通过超集JX(F)研究X(F)的可能性。这项研究产生了莫德尔-韦尔定理，指出JX(F)是有限生成的，这个结果随后被推广到任意阿贝尔变体。Weil希望证明几何交X∩JX(F)是有限的，从而证明莫德尔猜想。然而，JX(F)的阿贝尔性质本身是一个有用的性质，但在应用于x的算术运算时，却被证明是一个障碍，而不是一个帮助。然而，雅可比矩阵随后被西格尔用来证明仿射曲线上整数点在数域上的有限性，从而使算术学家相信这个抽象结构的实用性。后来，Weil试图超越阿贝尔框架，考虑秩为n / X的向量束的模空间Bunn(X)[24]。Serre[20]在他为Weil写的讣告中把这项工作描述为“作为分析呈现的文本，其意义本质上是代数的，但其动机是算术的。”他正确地强调了这篇论文的幻想性质，这篇论文写于几何不变量理论出现之前，而几何不变量理论的出现使得系统地处理这种模空间成为可能。今天，它们在1991年数学学科分类的各种几何版本中发挥着重要作用。由EPSRC拨款EP/M024830/1资助。C©0000(版权持有人)

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引用次数: 2

On categories of (𝜑,Γ)-modules 关于(变量，Γ)-模的范畴

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/pspum/097.2/01707

K. Kedlaya, Jonathan Pottharst

引用次数: 0

Singular Hermitian metrics and positivity of direct images of pluricanonical bundles 奇异厄米度量和多音束直接像的正性

Algebraic Geometry: Salt Lake City 2015

Pub Date : 2018-06-01 DOI: 10.1090/PSPUM/097.1/01684

Mihai Păun

This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.

这是一篇说明性文章。在第一部分中，我们回顾了关于无扭相干轴上奇异厄米度量的定义和一些结果。它们为研究扭曲多音束的直接像的性质提供了完美的平台，我们将在第二部分进行调查。

引用次数: 27

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