Algebraic Geometry: Salt Lake City 2015最新文献

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

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

{"title":"Θ-stratifications, Θ-reductive stacks, and\u0000 applications","authors":"Daniel Halpern-Leistner","doi":"10.1090/PSPUM/097.1/01678","DOIUrl":"https://doi.org/10.1090/PSPUM/097.1/01678","url":null,"abstract":"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 ","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127246140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

{"title":"Principal bundles and reciprocity laws in\u0000 number theory","authors":"Minhyong Kim","doi":"10.1090/PSPUM/097.2/01708","DOIUrl":"https://doi.org/10.1090/PSPUM/097.2/01708","url":null,"abstract":"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)","PeriodicalId":412716,"journal":{"name":"Algebraic Geometry: Salt Lake City\n 2015","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128148597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

引用次数: 27