Representation Stability, Configuration Spaces, and Deligne
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Nori Fundamental Gerbe of Tame Stacks 3
THE NORI FUNDAMENTAL GERBE OF TAME STACKS INDRANIL BISWAS AND NIELS BORNE Abstract. Given an algebraic stack, we compare its Nori fundamental group with that of its coarse moduli space. We also study conditions under which the stack can be uniformized by an algebraic space. 1. Introduction The aim here is to show that the results of [Noo04] concerning the ´etale fundamental group of algebraic stacks also hold for the Nori fundamental group. Let us start by recalling Noohi’s approach. Given a connected algebraic stack X, and a geometric point x : Spec Ω −→ X, Noohi generalizes the definition of Grothendieck’s ´etale fundamental group to get a profinite group π1(X, x) which classifies finite ´etale representable morphisms (coverings) to X. He then highlights a new feature specific to the stacky situation: for each geometric point x, there is a morphism ωx : Aut x −→ π1(X, x). Noohi first studies the situation where X admits a moduli space Y , and proceeds to show that if N is the closed normal subgroup of π1(X, x) generated by the images of ωx, for x varying in all geometric points, then π1(X, x) ≃ π1(Y,y) . N Noohi turns next to the issue of uniformizing algebraic stacks: he defines a Noetherian algebraic stack X as uniformizable if it admits a covering, in the above sense, that is an algebraic space. His main result is that this happens precisely when X is a Deligne– Mumford stack and for any geometric point x, the morphism ωx is injective. For our purpose, it turns out to be more convenient to use the Nori fundamental gerbe arXiv:1502.07023v3 [math.AG] 9 Sep 2015 defined in [BV12]. -
Arxiv:1601.00302V2 [Math.AG] 31 Oct 2019 Obntra Rbe,Idpneto Hrceitc N[ in Characteristic
UNIVERSAL STACKY SEMISTABLE REDUCTION SAM MOLCHO Abstract. Given a log smooth morphism f : X → S of toroidal embeddings, we perform a Raynaud-Gruson type operation on f to make it flat and with reduced fibers. We do this by studying the geometry of the associated map of cone complexes C(X) → C(S). As a consequence, we show that the toroidal part of semistable reduction of Abramovich-Karu can be done in a canonical way. 1. Introduction The semistable reduction theorem of [KKMSD73] is one of the foundations of the study of compactifications of moduli problems. Roughly, the main result of [KKMSD73] is that given a flat family X → S = Spec R over a discrete valua- tion ring, smooth over the generic point of R, there exists a finite base change ′ ′ ′ Spec R → Spec R and a modification X of the fiber product X ×Spec R Spec R such that the central fiber of X′ is a divisor with normal crossings which is reduced. Extensions of this result to the case where the base of X → S has higher di- mension are explored in the work [AK00] of Abramovich and Karu. Over a higher dimensional base, semistable reduction in the strictest sense is not possible; never- theless, the authors prove a version of the result, which they call weak semistable reduction. The strongest possible version of semistable reduction for a higher di- mensional base was proven recently in [ALT18]. In all cases, the statement is proven in two steps. In the first, one reduces to the case where X → S is toroidal. -
CHARACTERIZING ALGEBRAIC STACKS 1. Introduction Stacks
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 4, April 2008, Pages 1465–1476 S 0002-9939(07)08832-6 Article electronically published on December 6, 2007 CHARACTERIZING ALGEBRAIC STACKS SHARON HOLLANDER (Communicated by Paul Goerss) Abstract. We extend the notion of algebraic stack to an arbitrary subcanon- ical site C. If the topology on C is local on the target and satisfies descent for morphisms, we show that algebraic stacks are precisely those which are weakly equivalent to representable presheaves of groupoids whose domain map is a cover. This leads naturally to a definition of algebraic n-stacks. We also compare different sites naturally associated to a stack. 1. Introduction Stacks arise naturally in the study of moduli problems in geometry. They were in- troduced by Giraud [Gi] and Grothendieck and were used by Deligne and Mumford [DM] to study the moduli spaces of curves. They have recently become important also in differential geometry [Bry] and homotopy theory [G]. Higher-order general- izations of stacks are also receiving much attention from algebraic geometers and homotopy theorists. In this paper, we continue the study of stacks from the point of view of homotopy theory started in [H, H2]. The aim of these papers is to show that many properties of stacks and classical constructions with stacks are homotopy theoretic in nature. This homotopy-theoretical understanding gives rise to a simpler and more powerful general theory. In [H] we introduced model category structures on different am- bient categories in which stacks are the fibrant objects and showed that they are all Quillen equivalent. -
The Orbifold Chow Ring of Toric Deligne-Mumford Stacks
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 18, Number 1, Pages 193–215 S 0894-0347(04)00471-0 Article electronically published on November 3, 2004 THE ORBIFOLD CHOW RING OF TORIC DELIGNE-MUMFORD STACKS LEVA.BORISOV,LINDACHEN,ANDGREGORYG.SMITH 1. Introduction The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring in- troduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge num- bers, of the underlying variety. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [28], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and G¨ottsche [14] and Uribe [25] verify this conjecture when the orbifold is Symn(S)whereS is a smooth projective surface n with KS = 0 and the resolution is Hilb (S). The initial motivation for this project was to compare the orbifold Chow ring of a simplicial toric variety with the Chow ring of a crepant resolution. To achieve this goal, we first develop the theory of toric Deligne-Mumford stacks. Modeled on simplicial toric varieties, a toric Deligne-Mumford stack corresponds to a combinatorial object called a stacky fan. As a first approximation, this object is a simplicial fan with a distinguished lattice point on each ray in the fan. -
MOTIVIC VOLUMES of FIBERS of TROPICALIZATION 3 the Class of Y in MX , and We Endow MX with the Topology Given by the Dimension filtration
MOTIVIC VOLUMES OF FIBERS OF TROPICALIZATION JEREMY USATINE Abstract. Let T be an algebraic torus over an algebraically closed field, let X be a smooth closed subvariety of a T -toric variety such that U = X ∩ T is not empty, and let L (X) be the arc scheme of X. We define a tropicalization map on L (X) \ L (X \ U), the set of arcs of X that do not factor through X \ U. We show that each fiber of this tropicalization map is a constructible subset of L (X) and therefore has a motivic volume. We prove that if U has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev’s theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser’s motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization. 1. Introduction and Statements of Main Results Let k be an algebraically closed field, let T be an algebraic torus over k with character lattice M, and let X be a smooth closed subvariety of a T -toric variety such that U = X ∩ T 6= ∅. Let L (X) be the arc scheme of X. In this paper, we define a tropicalization map trop : L (X) \ L (X \ U) → M ∨ on the subset of arcs that do not factor through X \ U. -
Extension by Conservation. Sikorski's Theorem
Logical Methods in Computer Science Vol. 14(4:8)2018, pp. 1–17 Submitted Dec. 23, 2016 https://lmcs.episciences.org/ Published Oct. 30, 2018 EXTENSION BY CONSERVATION. SIKORSKI'S THEOREM DAVIDE RINALDI AND DANIEL WESSEL Universit`adi Verona, Dipartimento di Informatica, Strada le Grazie 15, 37134 Verona, Italy e-mail address: [email protected] e-mail address: [email protected] Abstract. Constructive meaning is given to the assertion that every finite Boolean algebra is an injective object in the category of distributive lattices. To this end, we employ Scott's notion of entailment relation, in which context we describe Sikorski's extension theorem for finite Boolean algebras and turn it into a syntactical conservation result. As a by-product, we can facilitate proofs of related classical principles. 1. Introduction Due to a time-honoured result by Sikorski (see, e.g., [36, x33] and [18]), the injective objects in the category of Boolean algebras have been identified precisely as the complete Boolean algebras. In other words, a Boolean algebra C is complete if and only if, for every morphism f : B ! C of Boolean algebras, where B is a subalgebra of B0, there is an extension g : B0 ! C of f onto B0. More generally, it has later been shown by Balbes [3], and Banaschewski and Bruns [5], that a distributive lattice is an injective object in the category of distributive lattices if and only if it is a complete Boolean algebra. A popular proof of Sikorski's theorem proceeds as follows: by Zorn's lemma the given morphism on B has a maximal partial extension, which by a clever one-step extension principle [18, 7] can be shown to be total on B0. -
Handbook of Moduli
Equivariant geometry and the cohomology of the moduli space of curves Dan Edidin Abstract. In this chapter we give a categorical definition of the integral cohomology ring of a stack. For quotient stacks [X=G] the categorical co- ∗ homology ring may be identified with the equivariant cohomology HG(X). Identifying the stack cohomology ring with equivariant cohomology allows us to prove that the cohomology ring of a quotient Deligne-Mumford stack is rationally isomorphic to the cohomology ring of its coarse moduli space. The theory is presented with a focus on the stacks Mg and Mg of smooth and stable curves respectively. Contents 1 Introduction2 2 Categories fibred in groupoids (CFGs)3 2.1 CFGs of curves4 2.2 Representable CFGs5 2.3 Curves and quotient CFGs6 2.4 Fiber products of CFGs and universal curves 10 3 Cohomology of CFGs and equivariant cohomology 13 3.1 Motivation and definition 14 3.2 Quotient CFGs and equivariant cohomology 16 3.3 Equivariant Chow groups 19 3.4 Equivariant cohomology and the CFG of curves 21 4 Stacks, moduli spaces and cohomology 22 4.1 Deligne-Mumford stacks 23 4.2 Coarse moduli spaces of Deligne-Mumford stacks 27 4.3 Cohomology of Deligne-Mumford stacks and their moduli spaces 29 2000 Mathematics Subject Classification. Primary 14D23, 14H10; Secondary 14C15, 55N91. Key words and phrases. stacks, moduli of curves, equivariant cohomology. The author was supported by NSA grant H98230-08-1-0059 while preparing this article. 2 Cohomology of the moduli space of curves 1. Introduction The study of the cohomology of the moduli space of curves has been a very rich research area for the last 30 years. -
Introduction to Algebraic Stacks
Introduction to Algebraic Stacks K. Behrend December 17, 2012 Abstract These are lecture notes based on a short course on stacks given at the Newton Institute in Cambridge in January 2011. They form a self-contained introduction to some of the basic ideas of stack theory. Contents Introduction 3 1 Topological stacks: Triangles 8 1.1 Families and their symmetry groupoids . 8 Various types of symmetry groupoids . 9 1.2 Continuous families . 12 Gluing families . 13 1.3 Classification . 15 Pulling back families . 17 The moduli map of a family . 18 Fine moduli spaces . 18 Coarse moduli spaces . 21 1.4 Scalene triangles . 22 1.5 Isosceles triangles . 23 1.6 Equilateral triangles . 25 1.7 Oriented triangles . 25 Non-equilateral oriented triangles . 30 1.8 Stacks . 31 1.9 Versal families . 33 Generalized moduli maps . 35 Reconstructing a family from its generalized moduli map . 37 Versal families: definition . 38 Isosceles triangles . 40 Equilateral triangles . 40 Oriented triangles . 41 1 1.10 Degenerate triangles . 42 Lengths of sides viewpoint . 43 Embedded viewpoint . 46 Complex viewpoint . 52 Oriented degenerate triangles . 54 1.11 Change of versal family . 57 Oriented triangles by projecting equilateral ones . 57 Comparison . 61 The comparison theorem . 61 1.12 Weierstrass compactification . 64 The j-plane . 69 2 Formalism 73 2.1 Objects in continuous families: Categories fibered in groupoids 73 Fibered products of groupoid fibrations . 77 2.2 Families characterized locally: Prestacks . 78 Versal families . 79 2.3 Families which can be glued: Stacks . 80 2.4 Topological stacks . 81 Topological groupoids . 81 Generalized moduli maps: Groupoid Torsors . -
Algebraic Stacks Many of the Geometric Properties That Are Defined for Schemes (Smoothness, Irreducibility, Separatedness, Properness, Etc...)
Algebraic stacks Tom´as L. G´omez Tata Institute of Fundamental Research Homi Bhabha road, Mumbai 400 005 (India) [email protected] 9 November 1999 Abstract This is an expository article on the theory of algebraic stacks. After intro- ducing the general theory, we concentrate in the example of the moduli stack of vector budles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory. 1 Introduction The concept of algebraic stack is a generalization of the concept of scheme, in the same sense that the concept of scheme is a generalization of the concept of projective variety. In many moduli problems, the functor that we want to study is not representable by a scheme. In other words, there is no fine moduli space. Usually this is because the objects that we want to parametrize have automorphisms. But if we enlarge the category of schemes (following ideas that go back to Grothendieck and Giraud, and were developed by Deligne, Mumford and Artin) and consider algebraic stacks, then we can construct the “moduli stack”, that captures all the information that we would like in a fine moduli space. The idea of enlarging the category of algebraic varieties to study moduli problems is not new. In fact A. Weil invented the concept of abstract variety to give an algebraic arXiv:math/9911199v1 [math.AG] 25 Nov 1999 construction of the Jacobian of a curve. These notes are an introduction to the theory of algebraic stacks. I have tried to emphasize ideas and concepts through examples instead of detailed proofs (I give references where these can be found). -
The Algebraic Stack Mg
The algebraic stack Mg Raymond van Bommel 9 October 2015 The author would like to thank Bas Edixhoven, Jinbi Jin, David Holmes and Giulio Orecchia for their valuable contributions. 1 Fibre products and representability Recall from last week ([G]) the definition of categories fibred in groupoids over a category S. Besides objects there are 1-morphisms (functors) between objects and 2-morphisms (natural transformations (that are isomorphisms automatically)) between the 1-morphisms. They form a so-called 2-category, CFG. We also defined the fibre product in CFG. In this chapter we will take a general base category S unless otherwise indicated. However, if wanted, you can think of S as being Sch. Given an object S 2 Ob(S), we can consider it as category fibred in groupoids, which is also called S=S, by taking arrows T ! S in S as objects and arrows over S in S as morphisms. The functor S=S !S is the forgetful functor, forgetting the map to S. Exercise 1. Let S; T 2 Ob(S). Check that the fibre (S=S)(T ) is isomorphic to the set Hom(T;S), i.e. the category whose objects are the elements of Hom(T;S) and whose arrows are the identities. Proposition 2. Let X be a category fibred in groupoids over S. Then X is equivalent to an object of S (i.e. there is an equivalence of categories that commutes with the projection to S) if and only if X has a final object. Proof. Given an object S 2 Ob(S) the associated category fibred in groupoids has final object id: S ! S. -
Geometricity for Derived Categories of Algebraic Stacks 3
GEOMETRICITY FOR DERIVED CATEGORIES OF ALGEBRAIC STACKS DANIEL BERGH, VALERY A. LUNTS, AND OLAF M. SCHNURER¨ To Joseph Bernstein on the occasion of his 70th birthday Abstract. We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of trian- gulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of co- herent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field. Contents 1. Introduction 1 2. Preliminaries 4 3. Root constructions 8 4. Semiorthogonal decompositions for root stacks 12 5. Differential graded enhancements and geometricity 20 6. Geometricity for dg enhancements of algebraic stacks 25 Appendix A. Bounded derived category of coherent modules 28 References 29 1. Introduction The derived category of a variety or, more generally, of an algebraic stack, is arXiv:1601.04465v2 [math.AG] 22 Jul 2016 traditionally studied in the context of triangulated categories. Although triangu- lated categories are certainly powerful, they do have some shortcomings. Most notably, the category of triangulated categories seems to have no tensor product, no concept of duals, and categories of triangulated functors have no obvious trian- gulated structure. A remedy to these problems is to work instead with differential graded categories, also called dg categories. We follow this approach and replace the derived category Dpf (X) of perfect complexes on a variety or an algebraic stack dg X by a certain dg category Dpf (X) which enhances Dpf (X) in the sense that its homotopy category is equivalent to Dpf (X). -
Higher Analytic Stacks and Gaga Theorems
HIGHER ANALYTIC STACKS AND GAGA THEOREMS MAURO PORTA AND TONY YUE YU Abstract. We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert’s theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre’s GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-étale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special case. Contents 1. Introduction 2 2. Higher geometric stacks 4 2.1. Notations 4 2.2. Geometric contexts 5 2.3. Higher geometric stacks 6 2.4. Functorialities of the category of sheaves 8 2.5. Change of geometric contexts 16 3. Examples of higher geometric stacks 17 3.1. Higher algebraic stacks 17 3.2. Higher complex analytic stacks 17 3.3. Higher non-archimedean analytic stacks 18 3.4. Relative higher algebraic stacks 18 4. Proper morphisms of analytic stacks 19 5. Direct images of coherent sheaves 26 5.1. Sheaves on geometric stacks 26 arXiv:1412.5166v2 [math.AG] 28 Jul 2016 5.2. Coherence of derived direct images for algebraic stacks 32 5.3. Coherence of derived direct images for analytic stacks 34 6. Analytification functors 38 Date: December 16, 2014 (Revised on July 6, 2016). 2010 Mathematics Subject Classification. Primary 14A20; Secondary 14G22 32C35 14F05. Key words and phrases.