# Vector Bundles on Algebraic Varieties

Size: px
Start display at page:

Transcription

1 Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general topological spaces; we will only examine those that have the additional structure of an algebraic variety. 1 Construction of a vector bundle Fixing a field, the affine variety A n together with a point designated as the origin has a structure of the vector space n. Because we want to treat our vector spaces as algebraic varieties, each vector space is embedded as an affine variety inside another variety referred to as the total space, and is attached to the base space by means of a morphism of varieties. Definition. A family of vector spaces over an algebraic variety X is a surjective morphism of varieties : E X, together with a fixed vector space structure over every fiber E x = 1 (x): E x = A n n. It is usually required that the dimension n of the vector space be constant over all fibers and finite. Definition. A morphism between the two families of vector spaces : E X and ρ : F X is given by a commutative diagram E ϕ F X ρ 1

2 where the morphism of varieties ϕ : E F, when restricted to any fiber E x (x X), is required to be a linear map of E x into F x with respect to their vector space structures. Furthermore, if ϕ gives an isomorphism of vector spaces between E x and F x for all x X, then the diagram is an isomorphism. Any family of vector spaces : E X may be restricted to an open subset U X. Then the restriction is also a family of vector spaces. U : 1 (U) U Definition. The trivial bundle, a particularly nice family of vector spaces, is simply the direct product of a vector space and the variety together with the projection onto the variety: E = X A n, : E X, : (x, v) x. In taing the direct product of two varieties, we should discuss what sort of varieties we are woring with. We want X to possibly be projective, and A n is affine. In the absence of the full generality of schemes, we can tae our varieties to be quasiprojective, that is a closed subset of an open subset of a projective space P n. Notice that the class of quasiprojective varieties includes both affine and projective varieties. The direct product of two quasiprojective varieties, defined via the Segre embedding into a projective space, can be shown to be again a quasiprojective variety. Definition. A vector bundle is a locally trivial family of vector spaces. More precisely, a family : E X is a vector bundle if there exists an open cover X = U α such that the restriction to each U α is isomorphic to a trivial bundle via a diagram 1 (U α ) ϕ α U α A n U α ρ where ρ is the projection to the first coordinate and ϕ α is a morphism of varieties giving vector space isomorphisms over fibers of and ρ. In an abuse of notation, we sometimes conflate the vector bundle and the total space, E. 2

3 The (constant) dimension n of the vector spaces is the ran of the vector bundle. A ran-1 vector bundle is referred to as a line bundle, for the obvious reason. Fixing an open cover {U α } and set of isomorphisms {ϕ α }, we can thin of each ϕ α as giving coordinate systems for the fibers E x (x U α ), mapping them into A n. For any nonempty intersection V = U α U β, the diagram ϕ α V A n 1 (V ) V A n V ϕ β indicates that we have the change of coordinates ϕ 1 α ϕ β {x} A n E x {x} A n between any two coordinatizations of a fiber. It can be shown that the matrix of the change of coordinates has entries given by regular functions on the intersection V. 2 Example: the tautological bundle over P n Each point of a projective space is associated with a one-dimensional subspace of an affine space; this suggests a particular line bundle, nown as the tautological bundle. Given the projective space P n, our total space E is a subset of P n An+1, where E = {(p, v) p P n, v l p }, and l p denotes the one-dimensional subspace of A n+1 associated with p. The morphism : E P n is simply the projection to the first coordinate. To show that this is a vector bundle, it is sufficient to construct a local trivialization of E. Fix a coordinate system of E, where a point (p, v) E is (p, v) = ((p 0 :... : p n ), (v 0,..., v n )), and the p i are homogeneous coordinates. Then P n has the standard open cover {U i }, where U i is the affine subset with p i 0. For each U i, we want a morphism ϕ i such that the diagram 3

4 1 (U i ) ϕ i U i A 1 U i trivializes U i. The construction is straightforward. Define ϕ i as ϕ i : (p, v) (p, v i ) ϕ 1 i : (p, v i ) (p, (v i /p i )p). Over a fiber of a given point p P n, it is clear that ϕ i is a linear map. Also, because the affine point v is a scalar multiple of p by construction and p i 0, the inverse ϕ 1 recovers v. Finally, the projective component p is preserved by all morphisms, so the diagram commutes. 3 Sections of a vector bundle A vector bundle is often viewed as a sheaf of sections. Definition. A section of a vector bundle : E X is a morphism s : X E such that s = id. One can visualize a section as a slice of the total space for which the projection is bijective. The set of all sections over an open subset U X is denoted L E (U). Any two sections s 1, s 2 L E (U) admit a sum (s 1 + s 2 )(x) = s 1 (x) + s 2 (x), and a section s L E (U) admits scaling by a regular function f on U (fs)(x) = f(x)s(x) where f(x) acts as on the vector space E x. Then L E (U) is a module over the regular functions on U, O X (U). In fact, L E is a sheaf of modules over the structure sheaf O X. Furthermore, it can be shown that there is a one-to-one correspondence of vector bundles over X and locally free sheaves of O X -modules with finite ran, i.e. sheaves L for which L (U α ) = n i=1 O X on some open cover U α ; we will not prove this here. This formulation of vector bundles maes a number of constructions easier, such as direct sums, tensor products, and duals of bundles. 4

5 4 The tangent bundle and differential forms Recall that given an affine smooth variety X A n, each point p X has a tangent space T p X defined as T p X = {v A n (d p f)(v) = 0 for all f I(X)}, where the differential of a polynomial f at p X is the function on A n n ( ) f (d p f)(x) = (p) (x i p i ). x i i=1 The value of a differential (d p f)(x) is a polynomial in the coordinates of both p and x, so the space E = {(p, x) x T p X} X A n is an affine variety defined by the polynomials {f, df} for f I(X), and is a vector bundle nown as the tangent bundle. Verifying that E is locally trivial is a little tricy (and is easier with the sheaf of modules formulation of a vector bundle), so we do not do it here. The idea is that the tangent space at a point is a local property. The dual of the tangent bundle is the cotangent bundle Ω, the m th exterior power Ω m is the bundle of m-dimensional differential forms, and the highest exterior power Ω n is a line bundle nown as the canonical bundle. These vector bundles are of particular importance as they turn out to be intrinsically defined, i.e. they do not depend on any particular embedding. References [1] Shafarevich, Igor. Basic Algebraic Geometry. Springer-Verlag, [2] Smith, Karen E., et al. An Invitation to Algebraic Geometry. Springer- Verlag,

### Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,

Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.

### PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### Resolution of Singularities in Algebraic Varieties

Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.

### ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

### Vector Bundles vs. Jesko Hüttenhain. Spring Abstract

Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

### ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical

### AG NOTES MARK BLUNK. 3. Varieties

AG NOTES MARK BLUNK Abstract. Some rough notes to get you all started. 1. Introduction Here is a list of the main topics that are discussed and used in my research talk. The information is rough and brief,

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### 10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

### Introduction to Chiral Algebras

Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

### LECTURE 2. (TEXED): IN CLASS: PROBABLY LECTURE 3. MANIFOLDS 1. FALL TANGENT VECTORS.

LECTURE 2. (TEXED): IN CLASS: PROBABLY LECTURE 3. MANIFOLDS 1. FALL 2006. TANGENT VECTORS. Overview: Tangent vectors, spaces and bundles. First: to an embedded manifold of Euclidean space. Then to one

### PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What

### A Bridge between Algebra and Topology: Swan s Theorem

A Bridge between Algebra and Topology: Swan s Theorem Daniel Hudson Contents 1 Vector Bundles 1 2 Sections of Vector Bundles 3 3 Projective Modules 4 4 Swan s Theorem 5 Introduction Swan s Theorem is a

### Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### Meta-Theorem 1. Suppose K/k is a finite G-Galois extension. Let C K C k

Galois Descent and Severi-Brauer Varieties. Eric Brussel Emory University Throughout this paper / will be a finite field extension. The goal of Galois descent is to characterize objects defined over that

### COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES ROBERT M. GURALNICK AND B.A. SETHURAMAN Abstract. In this note, we show that the set of all commuting d-tuples of commuting n n matrices that

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### Algebraic Geometry I Lectures 14 and 15

Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results

### ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

### Homology and Cohomology of Stacks (Lecture 7)

Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic

### Basic Algebraic Geometry 1

Igor R. Shafarevich Basic Algebraic Geometry 1 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Table of Contents Volume 1

### fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

### Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

### J. Huisman. Abstract. let bxc be the fundamental Z=2Z-homology class of X. We show that

On the dual of a real analytic hypersurface J. Huisman Abstract Let f be an immersion of a compact connected smooth real analytic variety X of dimension n into real projective space P n+1 (R). We say that

### Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### The Hitchin map, local to global

The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### 1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3

Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces

### PREQUANTIZATION OF SYMPLECTIC SUPERMANIFOLDS

Ninth International Conference on Geometry, Integrability and Quantization June 8 13, 2007, Varna, Bulgaria Ivaïlo M. Mladenov, Editor SOFTEX, Sofia 2008, pp 301 307 Geometry, Integrability and Quantization

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

### Curtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups

Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider

### Math 535a Homework 5

Math 535a Homework 5 Due Monday, March 20, 2017 by 5 pm Please remember to write down your name on your assignment. 1. Let (E, π E ) and (F, π F ) be (smooth) vector bundles over a common base M. A vector

### ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### The Atiyah bundle and connections on a principal bundle

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### Fiberwise two-sided multiplications on homogeneous C*-algebras

Fiberwise two-sided multiplications on homogeneous C*-algebras Ilja Gogić Department of Mathematics University of Zagreb XX Geometrical Seminar Vrnjačka Banja, Serbia May 20 23, 2018 joint work with Richard

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### Math 396. Linear algebra operations on vector bundles

Math 396. Linear algebra operations on vector bundles 1. Motivation Let (X, O) be a C p premanifold with corners, 0 p. We have developed the notion of a C p vector bundle over X as a certain kind of C

### Motivic integration on Artin n-stacks

Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of

### arxiv:math/ v1 [math.ag] 18 Oct 2003

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI

### LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain

### Algebraic Geometry (Math 6130)

Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,

### which is a group homomorphism, such that if W V U, then

4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

### CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

### ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

### Symplectic and Poisson Manifolds

Symplectic and Poisson Manifolds Harry Smith In this survey we look at the basic definitions relating to symplectic manifolds and Poisson manifolds and consider different examples of these. We go on to

### SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

### Algebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?

Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials

### Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

### INTRODUCTION TO ALGEBRAIC GEOMETRY

INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1 Preliminary of Calculus on Manifolds 11 Tangent Vectors What are tangent vectors we encounter in Calculus? (1) Given a parametrised curve α(t) = ( x(t),

### Fibre Bundles: Trivial or Not?

Rijksuniversiteit Groningen Bachelor thesis in mathematics Fibre Bundles: Trivial or Not? Timon Sytse van der Berg Prof. Dr. Henk W. Broer (first supervisor) Prof. Dr. Jaap Top (second supervisor) July

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

### Summer Algebraic Geometry Seminar

Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties

### Quantizations and classical non-commutative non-associative algebras

Journal of Generalized Lie Theory and Applications Vol. (008), No., 35 44 Quantizations and classical non-commutative non-associative algebras Hilja Lisa HURU and Valentin LYCHAGIN Department of Mathematics,

### CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

### HOMOMORPHISMS OF VECTOR BUNDLES ON CURVES AND PARABOLIC VECTOR BUNDLES ON A SYMMETRIC PRODUCT

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 9, September 2012, Pages 3017 3024 S 0002-9939(2012)11227-4 Article electronically published on January 24, 2012 HOMOMORPHISMS OF VECTOR

### THE HITCHIN FIBRATION

THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus

### Algebraic geometry over quaternions

Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

### ALGEBRAIC GEOMETRY CAUCHER BIRKAR

ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasi-projective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction

### Lecture 4 Super Lie groups

Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

### Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman

Commun. Math. Phys. 211, 359 363 2000) Communications in Mathematical Physics Springer-Verlag 2000 Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Hélène snault 1, I-Hsun

### THE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx

THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two
The text provided is a summary of a more advanced mathematical topic, specifically vector bundles on algebraic varieties. The content is quite dense and assumes a strong background in algebraic geometry. Here is a brief overview of the key points:

1. The text defines vector bundles over algebraic varieties, which are geometric objects that generalize the notion of vector spaces attached to each point of a space. 

2. It discusses the construction of vector bundles, including the trivial bundle and the tautological bundle over projective space.

3. The text introduces the concept of sections of a vector bundle, which are essentially "smooth" choices of vectors over each point.

4. It mentions important examples like the tangent bundle and cotangent bundle of a variety.

5. The text touches on more advanced topics like locally free sheaves and their correspondence with vector bundles.

6. Overall, this appears to be a technical introduction to the theory of vector bundles in algebraic geometry, aimed at readers with a solid background in the field.

In terms of the two questions:

1. This text does not contain any explicit mathematical questions or problems to solve.

2. As there are no questions, there are also no answers provided.

So the appropriate classification would be:
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Vector Bundles on Algebraic Varieties

Size: px
Start display at page:

Transcription

1 Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general topological spaces; we will only examine those that have the additional structure of an algebraic variety. 1 Construction of a vector bundle Fixing a field, the affine variety A n together with a point designated as the origin has a structure of the vector space n. Because we want to treat our vector spaces as algebraic varieties, each vector space is embedded as an affine variety inside another variety referred to as the total space, and is attached to the base space by means of a morphism of varieties. Definition. A family of vector spaces over an algebraic variety X is a surjective morphism of varieties : E X, together with a fixed vector space structure over every fiber E x = 1 (x): E x = A n n. It is usually required that the dimension n of the vector space be constant over all fibers and finite. Definition. A morphism between the two families of vector spaces : E X and ρ : F X is given by a commutative diagram E ϕ F X ρ 1

2 where the morphism of varieties ϕ : E F, when restricted to any fiber E x (x X), is required to be a linear map of E x into F x with respect to their vector space structures. Furthermore, if ϕ gives an isomorphism of vector spaces between E x and F x for all x X, then the diagram is an isomorphism. Any family of vector spaces : E X may be restricted to an open subset U X. Then the restriction is also a family of vector spaces. U : 1 (U) U Definition. The trivial bundle, a particularly nice family of vector spaces, is simply the direct product of a vector space and the variety together with the projection onto the variety: E = X A n, : E X, : (x, v) x. In taing the direct product of two varieties, we should discuss what sort of varieties we are woring with. We want X to possibly be projective, and A n is affine. In the absence of the full generality of schemes, we can tae our varieties to be quasiprojective, that is a closed subset of an open subset of a projective space P n. Notice that the class of quasiprojective varieties includes both affine and projective varieties. The direct product of two quasiprojective varieties, defined via the Segre embedding into a projective space, can be shown to be again a quasiprojective variety. Definition. A vector bundle is a locally trivial family of vector spaces. More precisely, a family : E X is a vector bundle if there exists an open cover X = U α such that the restriction to each U α is isomorphic to a trivial bundle via a diagram 1 (U α ) ϕ α U α A n U α ρ where ρ is the projection to the first coordinate and ϕ α is a morphism of varieties giving vector space isomorphisms over fibers of and ρ. In an abuse of notation, we sometimes conflate the vector bundle and the total space, E. 2

3 The (constant) dimension n of the vector spaces is the ran of the vector bundle. A ran-1 vector bundle is referred to as a line bundle, for the obvious reason. Fixing an open cover {U α } and set of isomorphisms {ϕ α }, we can thin of each ϕ α as giving coordinate systems for the fibers E x (x U α ), mapping them into A n. For any nonempty intersection V = U α U β, the diagram ϕ α V A n 1 (V ) V A n V ϕ β indicates that we have the change of coordinates ϕ 1 α ϕ β {x} A n E x {x} A n between any two coordinatizations of a fiber. It can be shown that the matrix of the change of coordinates has entries given by regular functions on the intersection V. 2 Example: the tautological bundle over P n Each point of a projective space is associated with a one-dimensional subspace of an affine space; this suggests a particular line bundle, nown as the tautological bundle. Given the projective space P n, our total space E is a subset of P n An+1, where E = {(p, v) p P n, v l p }, and l p denotes the one-dimensional subspace of A n+1 associated with p. The morphism : E P n is simply the projection to the first coordinate. To show that this is a vector bundle, it is sufficient to construct a local trivialization of E. Fix a coordinate system of E, where a point (p, v) E is (p, v) = ((p 0 :... : p n ), (v 0,..., v n )), and the p i are homogeneous coordinates. Then P n has the standard open cover {U i }, where U i is the affine subset with p i 0. For each U i, we want a morphism ϕ i such that the diagram 3

4 1 (U i ) ϕ i U i A 1 U i trivializes U i. The construction is straightforward. Define ϕ i as ϕ i : (p, v) (p, v i ) ϕ 1 i : (p, v i ) (p, (v i /p i )p). Over a fiber of a given point p P n, it is clear that ϕ i is a linear map. Also, because the affine point v is a scalar multiple of p by construction and p i 0, the inverse ϕ 1 recovers v. Finally, the projective component p is preserved by all morphisms, so the diagram commutes. 3 Sections of a vector bundle A vector bundle is often viewed as a sheaf of sections. Definition. A section of a vector bundle : E X is a morphism s : X E such that s = id. One can visualize a section as a slice of the total space for which the projection is bijective. The set of all sections over an open subset U X is denoted L E (U). Any two sections s 1, s 2 L E (U) admit a sum (s 1 + s 2 )(x) = s 1 (x) + s 2 (x), and a section s L E (U) admits scaling by a regular function f on U (fs)(x) = f(x)s(x) where f(x) acts as on the vector space E x. Then L E (U) is a module over the regular functions on U, O X (U). In fact, L E is a sheaf of modules over the structure sheaf O X. Furthermore, it can be shown that there is a one-to-one correspondence of vector bundles over X and locally free sheaves of O X -modules with finite ran, i.e. sheaves L for which L (U α ) = n i=1 O X on some open cover U α ; we will not prove this here. This formulation of vector bundles maes a number of constructions easier, such as direct sums, tensor products, and duals of bundles. 4

5 4 The tangent bundle and differential forms Recall that given an affine smooth variety X A n, each point p X has a tangent space T p X defined as T p X = {v A n (d p f)(v) = 0 for all f I(X)}, where the differential of a polynomial f at p X is the function on A n n ( ) f (d p f)(x) = (p) (x i p i ). x i i=1 The value of a differential (d p f)(x) is a polynomial in the coordinates of both p and x, so the space E = {(p, x) x T p X} X A n is an affine variety defined by the polynomials {f, df} for f I(X), and is a vector bundle nown as the tangent bundle. Verifying that E is locally trivial is a little tricy (and is easier with the sheaf of modules formulation of a vector bundle), so we do not do it here. The idea is that the tangent space at a point is a local property. The dual of the tangent bundle is the cotangent bundle Ω, the m th exterior power Ω m is the bundle of m-dimensional differential forms, and the highest exterior power Ω n is a line bundle nown as the canonical bundle. These vector bundles are of particular importance as they turn out to be intrinsically defined, i.e. they do not depend on any particular embedding. References [1] Shafarevich, Igor. Basic Algebraic Geometry. Springer-Verlag, [2] Smith, Karen E., et al. An Invitation to Algebraic Geometry. Springer- Verlag,

### Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,

Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.

### PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### Resolution of Singularities in Algebraic Varieties

Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.

### ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

### Vector Bundles vs. Jesko Hüttenhain. Spring Abstract

Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

### ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical

### AG NOTES MARK BLUNK. 3. Varieties

AG NOTES MARK BLUNK Abstract. Some rough notes to get you all started. 1. Introduction Here is a list of the main topics that are discussed and used in my research talk. The information is rough and brief,

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### 10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

### Introduction to Chiral Algebras

Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

### LECTURE 2. (TEXED): IN CLASS: PROBABLY LECTURE 3. MANIFOLDS 1. FALL TANGENT VECTORS.

LECTURE 2. (TEXED): IN CLASS: PROBABLY LECTURE 3. MANIFOLDS 1. FALL 2006. TANGENT VECTORS. Overview: Tangent vectors, spaces and bundles. First: to an embedded manifold of Euclidean space. Then to one

### PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What

### A Bridge between Algebra and Topology: Swan s Theorem

A Bridge between Algebra and Topology: Swan s Theorem Daniel Hudson Contents 1 Vector Bundles 1 2 Sections of Vector Bundles 3 3 Projective Modules 4 4 Swan s Theorem 5 Introduction Swan s Theorem is a

### Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### Meta-Theorem 1. Suppose K/k is a finite G-Galois extension. Let C K C k

Galois Descent and Severi-Brauer Varieties. Eric Brussel Emory University Throughout this paper / will be a finite field extension. The goal of Galois descent is to characterize objects defined over that

### COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES ROBERT M. GURALNICK AND B.A. SETHURAMAN Abstract. In this note, we show that the set of all commuting d-tuples of commuting n n matrices that

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### Algebraic Geometry I Lectures 14 and 15

Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results

### ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

### Homology and Cohomology of Stacks (Lecture 7)

Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic

### Basic Algebraic Geometry 1

Igor R. Shafarevich Basic Algebraic Geometry 1 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Table of Contents Volume 1

### fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

### Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

### J. Huisman. Abstract. let bxc be the fundamental Z=2Z-homology class of X. We show that

On the dual of a real analytic hypersurface J. Huisman Abstract Let f be an immersion of a compact connected smooth real analytic variety X of dimension n into real projective space P n+1 (R). We say that

### Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### The Hitchin map, local to global

The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### 1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3

Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces

### PREQUANTIZATION OF SYMPLECTIC SUPERMANIFOLDS

Ninth International Conference on Geometry, Integrability and Quantization June 8 13, 2007, Varna, Bulgaria Ivaïlo M. Mladenov, Editor SOFTEX, Sofia 2008, pp 301 307 Geometry, Integrability and Quantization

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

### Curtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups

Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider

### Math 535a Homework 5

Math 535a Homework 5 Due Monday, March 20, 2017 by 5 pm Please remember to write down your name on your assignment. 1. Let (E, π E ) and (F, π F ) be (smooth) vector bundles over a common base M. A vector

### ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### The Atiyah bundle and connections on a principal bundle

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### Fiberwise two-sided multiplications on homogeneous C*-algebras

Fiberwise two-sided multiplications on homogeneous C*-algebras Ilja Gogić Department of Mathematics University of Zagreb XX Geometrical Seminar Vrnjačka Banja, Serbia May 20 23, 2018 joint work with Richard

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### Math 396. Linear algebra operations on vector bundles

Math 396. Linear algebra operations on vector bundles 1. Motivation Let (X, O) be a C p premanifold with corners, 0 p. We have developed the notion of a C p vector bundle over X as a certain kind of C

### Motivic integration on Artin n-stacks

Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of

### arxiv:math/ v1 [math.ag] 18 Oct 2003

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI

### LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain

### Algebraic Geometry (Math 6130)

Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,

### which is a group homomorphism, such that if W V U, then

4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

### CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

### ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

### Symplectic and Poisson Manifolds

Symplectic and Poisson Manifolds Harry Smith In this survey we look at the basic definitions relating to symplectic manifolds and Poisson manifolds and consider different examples of these. We go on to

### SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

### Algebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?

Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials

### Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

### INTRODUCTION TO ALGEBRAIC GEOMETRY

INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1 Preliminary of Calculus on Manifolds 11 Tangent Vectors What are tangent vectors we encounter in Calculus? (1) Given a parametrised curve α(t) = ( x(t),

### Fibre Bundles: Trivial or Not?

Rijksuniversiteit Groningen Bachelor thesis in mathematics Fibre Bundles: Trivial or Not? Timon Sytse van der Berg Prof. Dr. Henk W. Broer (first supervisor) Prof. Dr. Jaap Top (second supervisor) July

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

### Summer Algebraic Geometry Seminar

Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties

### Quantizations and classical non-commutative non-associative algebras

Journal of Generalized Lie Theory and Applications Vol. (008), No., 35 44 Quantizations and classical non-commutative non-associative algebras Hilja Lisa HURU and Valentin LYCHAGIN Department of Mathematics,

### CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

### HOMOMORPHISMS OF VECTOR BUNDLES ON CURVES AND PARABOLIC VECTOR BUNDLES ON A SYMMETRIC PRODUCT

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 9, September 2012, Pages 3017 3024 S 0002-9939(2012)11227-4 Article electronically published on January 24, 2012 HOMOMORPHISMS OF VECTOR

### THE HITCHIN FIBRATION

THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus

### Algebraic geometry over quaternions

Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

### ALGEBRAIC GEOMETRY CAUCHER BIRKAR

ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasi-projective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction

### Lecture 4 Super Lie groups

Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

### Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman

Commun. Math. Phys. 211, 359 363 2000) Communications in Mathematical Physics Springer-Verlag 2000 Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Hélène snault 1, I-Hsun

### THE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx

THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two
user
The text provided is a summary of a more advanced mathematical topic, specifically vector bundles on algebraic varieties. The content is quite dense and assumes a strong background in algebraic geometry. Here is a brief overview of the key points:

1. The text defines vector bundles over algebraic varieties, which are geometric objects that generalize the notion of vector spaces attached to each point of a space. 

2. It discusses the construction of vector bundles, including the trivial bundle and the tautological bundle over projective space.

3. The text introduces the concept of sections of a vector bundle, which are essentially "smooth" choices of vectors over each point.

4. It mentions important examples like the tangent bundle and cotangent bundle of a variety.

5. The text touches on more advanced topics like locally free sheaves and their correspondence with vector bundles.

6. Overall, this appears to be a technical introduction to the theory of vector bundles in algebraic geometry, aimed at readers with a solid background in the field.

In terms of the two questions:

1. This text does not contain any explicit mathematical questions or problems to solve.

2. As there are no questions, there are also no answers provided.

So the appropriate classification would be:

TYPE: NO_QUESTION