'Cycle Groups for Artin Stacks'
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Families of Cycles and the Chow Scheme
Families of cycles and the Chow scheme DAVID RYDH Doctoral Thesis Stockholm, Sweden 2008 TRITA-MAT-08-MA-06 ISSN 1401-2278 KTH Matematik ISRN KTH/MAT/DA 08/05-SE SE-100 44 Stockholm ISBN 978-91-7178-999-0 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik måndagen den 11 augusti 2008 klockan 13.00 i Nya kollegiesalen, F3, Kungl Tek- niska högskolan, Lindstedtsvägen 26, Stockholm. © David Rydh, maj 2008 Tryck: Universitetsservice US AB iii Abstract The objects studied in this thesis are families of cycles on schemes. A space — the Chow variety — parameterizing effective equidimensional cycles was constructed by Chow and van der Waerden in the first half of the twentieth century. Even though cycles are simple objects, the Chow variety is a rather intractable object. In particular, a good func- torial description of this space is missing. Consequently, descriptions of the corresponding families and the infinitesimal structure are incomplete. Moreover, the Chow variety is not intrinsic but has the unpleasant property that it depends on a given projective embedding. A main objective of this thesis is to construct a closely related space which has a good functorial description. This is partly accomplished in the last paper. The first three papers are concerned with families of zero-cycles. In the first paper, a functor parameterizing zero-cycles is defined and it is shown that this functor is represented by a scheme — the scheme of divided powers. This scheme is closely related to the symmetric product. -
Arxiv:2006.16553V2 [Math.AG] 26 Jul 2020
ON ULRICH BUNDLES ON PROJECTIVE BUNDLES ANDREAS HOCHENEGGER Abstract. In this article, the existence of Ulrich bundles on projective bundles P(E) → X is discussed. In the case, that the base variety X is a curve or surface, a close relationship between Ulrich bundles on X and those on P(E) is established for specific polarisations. This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and some surfaces. 1. Introduction Given a smooth projective variety X, polarised by a very ample divisor A, let i: X ֒→ PN be the associated closed embedding. A locally free sheaf F on X is called Ulrich bundle (with respect to A) if and only if it satisfies one of the following conditions: • There is a linear resolution of F: ⊕bc ⊕bc−1 ⊕b0 0 → OPN (−c) → OPN (−c + 1) →···→OPN → i∗F → 0, where c is the codimension of X in PN . • The cohomology H•(X, F(−pA)) vanishes for 1 ≤ p ≤ dim(X). • For any finite linear projection π : X → Pdim(X), the locally free sheaf π∗F splits into a direct sum of OPdim(X) . Actually, by [18], these three conditions are equivalent. One guiding question about Ulrich bundles is whether a given variety admits an Ulrich bundle of low rank. The existence of such a locally free sheaf has surprisingly strong implications about the geometry of the variety, see the excellent surveys [6, 14]. Given a projective bundle π : P(E) → X, this article deals with the ques- tion, what is the relation between Ulrich bundles on the base X and those on P(E)? Note that answers to such a question depend much on the choice arXiv:2006.16553v3 [math.AG] 15 Aug 2021 of a very ample divisor. -
The Cones of Effective Cycles on Projective Bundles Over Curves
The cones of effective cycles on projective bundles over curves Mihai Fulger 1 Introduction The study of the cones of curves or divisors on complete varieties is a classical subject in Algebraic Geometry (cf. [10], [9], [4]) and it still is an active research topic (cf. [1], [11] or [2]). However, little is known if we pass to higher (co)dimension. In this paper we study this problem in the case of projective bundles over curves and describe the cones of effective cycles in terms of the numerical data appearing in a Harder–Narasimhan filtration. This generalizes to higher codimension results of Miyaoka and others ([15], [3]) for the case of divisors. An application to projective bundles over a smooth base of arbitrary dimension is also given. Given a smooth complex projective variety X of dimension n, consider the real vector spaces The real span of {[Y ] : Y is a subvariety of X of codimension k} N k(X) := . numerical equivalence of cycles We also denote it by Nn−k(X) when we work with dimension instead of codimension. The direct sum n k N(X) := M N (X) k=0 is a graded R-algebra with multiplication induced by the intersection form. i Define the cones Pseff (X) = Pseffn−i(X) as the closures of the cones of effective cycles in i N (X). The elements of Pseffi(X) are usually called pseudoeffective. Dually, we have the nef cones k k Nef (X) := {α ∈ Pseff (X)| α · β ≥ 0 ∀β ∈ Pseffk(X)}. Now let C be a smooth complex projective curve and let E be a locally free sheaf on C of rank n, deg E degree d and slope µ(E) := rankE , or µ for short. -
Algebraic Cycles and Fibrations
Documenta Math. 1521 Algebraic Cycles and Fibrations Charles Vial Received: March 23, 2012 Revised: July 19, 2013 Communicated by Alexander Merkurjev Abstract. Let f : X → B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f. One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is “built” from the motives of varieties of dimension less than the dimension of B. 2010 Mathematics Subject Classification: 14C15, 14C25, 14C05, 14D99 Keywords and Phrases: Algebraic cycles, Chow groups, quadric bun- dles, motives, Chow–K¨unneth decomposition. Contents 1. Surjective morphisms and motives 1526 2. OntheChowgroupsofthefibres 1530 3. A generalisation of the projective bundle formula 1532 4. On the motive of quadric bundles 1534 5. Onthemotiveofasmoothblow-up 1536 6. Chow groups of varieties fibred by varieties with small Chow groups 1540 7. Applications 1547 References 1551 Documenta Mathematica 18 (2013) 1521–1553 1522 Charles Vial For a scheme X over a field k, CHi(X) denotes the rational Chow group of i- dimensional cycles on X modulo rational equivalence. Throughout, f : X → B will be a projective surjective morphism defined over k from a quasi-projective variety X of dimension dX to an irreducible quasi-projective variety B of di- mension dB, with various extra assumptions which will be explicitly stated. Let h be the class of a hyperplane section in the Picard group of X. -
Equivariant Operational Chow Rings of T-Linear Schemes
EQUIVARIANT OPERATIONAL CHOW RINGS OF T-LINEAR SCHEMES RICHARD P. GONZALES * Abstract. We study T -linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a K¨unnethformula for the equivariant Chow groups of these schemes. Using such formula, we show that equivariant Kronecker duality holds for the equivariant operational Chow rings (or equivariant Chow cohomology) of T -linear schemes. As an application, we obtain a presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties. 1. Introduction and motivation Let G be a connected reductive group defined over an algebraically closed field k of characteristic zero. Let B be a Borel subgroup of G and T ⊂ B be a maximal torus of G. An algebraic variety X, equipped with an action of G, is spherical if it contains a dense orbit of B. (Usually spherical varieties are assumed to be normal but this condition is not needed here.) Spherical varieties have been extensively studied in the works of Akhiezer, Brion, Knop, Luna, Pauer, Vinberg, Vust and others. For an up-to-date discussion of spherical varieties, as well as a comprehensive bibliography, see [Ti] and the references therein. If X is spherical, then it has a finite number of B- orbits, and thus, also a finite number of G-orbits (see e.g. [Vin], [Kn2]). In particular, T acts on X with a finite number of fixed points. These properties make spherical varieties particularly well suited for applying the methods of Goresky-Kottwitz-MacPherson [GKM], nowadays called GKM theory, in the topological setup, and Brion's extension of GKM theory [Br3] to the algebraic setting of equivariant Chow groups, as defined by Totaro, Edidin and Graham [EG-1]. -
Algebraic Cycles, Chow Varieties, and Lawson Homology Compositio Mathematica, Tome 77, No 1 (1991), P
COMPOSITIO MATHEMATICA ERIC M. FRIEDLANDER Algebraic cycles, Chow varieties, and Lawson homology Compositio Mathematica, tome 77, no 1 (1991), p. 55-93 <http://www.numdam.org/item?id=CM_1991__77_1_55_0> © Foundation Compositio Mathematica, 1991, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 77: 55-93,55 1991. (Ç) 1991 Kluwer Academic Publishers. Printed in the Netherlands. Algebraic cycles, Chow varieties, and Lawson homology ERIC M. FRIEDLANDER* Department of Mathematics, Northwestern University, Evanston, Il. 60208, U.S.A. Received 22 August 1989; accepted in revised form 14 February 1990 Following the foundamental work of H. Blaine Lawson [ 19], [20], we introduce new invariants for projective algebraic varieties which we call Lawson ho- mology groups. These groups are a hybrid of algebraic geometry and algebraic topology: the 1-adic Lawson homology group LrH2,+i(X, Zi) of a projective variety X for a given prime1 invertible in OX can be naively viewed as the group of homotopy classes of S’*-parametrized families of r-dimensional algebraic cycles on X. Lawson homology groups are covariantly functorial, as homology groups should be, and admit Galois actions. If i = 0, then LrH 2r+i(X, Zl) is the group of algebraic equivalence classes of r-cycles; if r = 0, then LrH2r+i(X , Zl) is 1-adic etale homology. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
Stability and Extremal Metrics on Toric and Projective Bundles
STABILITY AND EXTREMAL METRICS ON TORIC AND PROJECTIVE BUNDLES VESTISLAV APOSTOLOV, DAVID M. J. CALDERBANK, PAUL GAUDUCHON, AND CHRISTINA W. TØNNESEN-FRIEDMAN Abstract. This survey on extremal K¨ahlermetrics is a synthesis of several recent works—of G. Sz´ekelyhidi [39, 40], of Ross–Thomas [35, 36], and of our- selves [5, 6, 7]—but embedded in a new framework for studying extremal K¨ahler metrics on toric bundles following Donaldson [13] and Szekelyhidi [40]. These works build on ideas Donaldson, Tian, and many others concerning stability conditions for the existence of extremal K¨ahler metrics. In particular, we study notions of relative slope stability and relative uniform stability and present ex- amples which suggest that these notions are closely related to the existence of extremal K¨ahlermetrics. Contents 1. K-stability for extremal K¨ahlermetrics 2 1.1. Introduction to stability 2 1.2. Finite dimensional motivation 2 1.3. The infinite dimensional analogue 3 1.4. Test configurations 3 1.5. The modified Futaki invariant 4 1.6. K-stability 5 2. Rigid toric bundles with semisimple base 6 2.1. The rigid Ansatz 6 2.2. The semisimple case 8 2.3. The isometry Lie algebra in the semisimple case 8 2.4. The extremal vector field 9 2.5. K-energy and the Futaki invariant 9 2.6. Stability under small perturbations 10 2.7. Existence? 14 2.8. The homogeneous formalism and projective invariance 15 2.9. Examples 16 3. K-stability for rigid toric bundles 19 3.1. Toric test configurations 19 3.2. -
Arxiv:1409.5404V2 [Math.AG]
MOTIVIC CLASSES OF SOME CLASSIFYING STACKS DANIEL BERGH Abstract. We prove that the class of the classifying stack B PGLn is the multiplicative inverse of the class of the projective linear group PGLn in the Grothendieck ring of stacks K0(Stackk) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is known that the multiplicativity relation {T } = {S} · {PGLn} does not hold for all PGLn- torsors T → S, it holds for the universal PGLn-torsors for said n. Introduction Recall that the Grothendieck ring of varieties over a field k is defined as the free group on isomorphism classes {X} of varieties X subject to the scissors relations {X} = {X \ Z} + {Z} for closed subvarieties Z ⊂ X. We denote this ring by K0(Vark). Its multiplicative structure is induced by {X}·{Y } = {X × Y } for varieties X and Y . A fundamental question regarding K0(Vark) is for which groups G the multi- plicativity relation {T } = {G}·{S} holds for all G-torsors T → S. If G is special, in the sense of Serre and Grothendieck, this relation always holds. However, it was shown by Ekedahl that if G is a non-special connected linear group, then there exists a G-torsor T → S for which {T } 6= {G}·{S} in the ring K0(VarC) and its completion K0(VarC) [12, Theorem 2.2]. One can also construct a Grothendieck ring K (Stack ) for the larger 2-category b 0 k of algebraic stacks. We will recall its precise definition in Section 2. -
K3 Surfaces and String Duality
RU-96-98 hep-th/9611137 November 1996 K3 Surfaces and String Duality Paul S. Aspinwall Dept. of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855 ABSTRACT The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We arXiv:hep-th/9611137v5 7 Sep 1999 review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E E heterotic string, 8 × 8 and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. Contents 1 Introduction 2 2 Classical Geometry 4 2.1 Definition ..................................... 4 2.2 Holonomy ..................................... 7 2.3 Moduli space of complex structures . ..... 9 2.4 Einsteinmetrics................................. 12 2.5 AlgebraicK3Surfaces ............................. 15 2.6 Orbifoldsandblow-ups. .. 17 3 The World-Sheet Perspective 25 3.1 TheNonlinearSigmaModel . 25 3.2 TheTeichm¨ullerspace . .. 27 3.3 Thegeometricsymmetries . .. 29 3.4 Mirrorsymmetry ................................. 31 3.5 Conformalfieldtheoryonatorus . ... 33 4 Type II String Theory 37 4.1 Target space supergravity and compactification . -
Intersection Theory on Algebraic Stacks and Q-Varieties
Journal of Pure and Applied Algebra 34 (1984) 193-240 193 North-Holland INTERSECTION THEORY ON ALGEBRAIC STACKS AND Q-VARIETIES Henri GILLET Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Communicated by E.M. Friedlander Received 5 February 1984 Revised 18 May 1984 Contents Introduction ................................................................ 193 1. Preliminaries ................................................................ 196 2. Algebraic groupoids .......................................................... 198 3. Algebraic stacks ............................................................. 207 4. Chow groups of stacks ....................................................... 211 5. Descent theorems for rational K-theory. ........................................ 2 16 6. Intersection theory on stacks .................................................. 219 7. K-theory of stacks ........................................................... 229 8. Chcrn classes and the Riemann-Roth theorem for algebraic spaces ................ 233 9. comparison with Mumford’s product .......................................... 235 References .................................................................. 239 0. Introduction In his article 1161, Mumford constructed an intersection product on the Chow groups (with rational coefficients) of the moduli space . fg of stable curves of genus g over a field k of characteristic zero. Having such a product is important in study- ing the enumerative geometry of curves, and it is reasonable -
Intersection Theory
William Fulton Intersection Theory Second Edition Springer Contents Introduction. Chapter 1. Rational Equivalence 6 Summary 6 1.1 Notation and Conventions 6 .2 Orders of Zeros and Poles 8 .3 Cycles and Rational Equivalence 10 .4 Push-forward of Cycles 11 .5 Cycles of Subschemes 15 .6 Alternate Definition of Rational Equivalence 15 .7 Flat Pull-back of Cycles 18 .8 An Exact Sequence 21 1.9 Affine Bundles 22 1.10 Exterior Products 24 Notes and References 25 Chapter 2. Divisors 28 Summary 28 2.1 Cartier Divisors and Weil Divisors 29 2.2 Line Bundles and Pseudo-divisors 31 2.3 Intersecting with Divisors • . 33 2.4 Commutativity of Intersection Classes 35 2.5 Chern Class of a Line Bundle 41 2.6 Gysin Map for Divisors 43 Notes and References 45 Chapter 3. Vector Bundles and Chern Classes 47 Summary 47 3.1 Segre Classes of Vector Bundles 47 3.2 Chern Classes . ' 50 • 3.3 Rational Equivalence on Bundles 64 Notes and References 68 Chapter 4. Cones and Segre Classes 70 Summary . 70 . 4.1 Segre Class of a Cone . 70 X Contents 4.2 Segre Class of a Subscheme 73 4.3 Multiplicity Along a Subvariety 79 4.4 Linear Systems 82 Notes and References 85 Chapter 5. Deformation to the Normal Cone 86 Summary 86 5.1 The Deformation 86 5.2 Specialization to the Normal Cone 89 Notes and References 90 Chapter 6. Intersection Products 92 Summary 92 6.1 The Basic Construction 93 6.2 Refined Gysin Homomorphisms 97 6.3 Excess Intersection Formula 102 6.4 Commutativity 106 6.5 Functoriality 108 6.6 Local Complete Intersection Morphisms 112 6.7 Monoidal Transforms 114 Notes and References 117 Chapter 7.