256B Algebraic Geometry
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Equivariant Geometric K-Homology with Coefficients
Equivariant geometric K-homology with coefficients Michael Walter Equivariant geometric K-homology with coefficients Diplomarbeit vorgelegt von Michael Walter geboren in Lahr angefertigt am Mathematischen Institut der Georg-August-Universität zu Göttingen 2010 v Equivariant geometric K-homology with coefficients Michael Walter Abstract K-homology is the dual of K-theory. Kasparov’s analytic version, where cycles are given by (ab- stract) elliptic operators over (not necessarily commutative) spaces, has proved to be an extremely powerful tool which, together with its bivariant generalization KK-theory, lies at the heart of many important results at the intersection of algebraic topology, functional analysis and geome- try. Independently, Baum and Douglas have proposed a geometric version of K-homology inspired by singular bordism. Cycles for this theory are given by vector bundles over compact Spinc- manifolds with boundary which map to the target, i.e. E f (M, BM) / (X, Y). There is a natural transformation to analytic K-homology defined by sending such a cycle to the pushforward of the class determined by the twisted Dirac operator. It is well-known to be an isomorphism, although a rigorous proof has appeared only recently. While both theories have obvious generalizations to the equivariant case and coefficients, the question whether these remain isomorphic is far from trivial (and has negative answer in the general case). In their work on equivariant correspondences Emerson and Meyer have isolated a useful sufficient condition for their theory which, while vastly more general, only deals with the absolute case. Our focus is not so much to construct a geometric theory in the most general situation, but to show that in the presence of a group action and coefficients the above picture still gives a generalized homology theory in a very geometrical way, isomorphic to Kasparov’s theory. -
Ribbons and Their Canonical Embeddings
transactions of the american mathematical society Volume 347, Number 3, March 1995 RIBBONS AND THEIR CANONICAL EMBEDDINGS DAVE BAYER AND DAVID EISENBUD Abstract. We study double structures on the projective line and on certain other varieties, with a view to having a nice family of degenerations of curves and K3 surfaces of given genus and Clifford index. Our main interest is in the canonical embeddings of these objects, with a view toward Green's Conjecture on the free resolutions of canonical curves. We give the canonical embeddings explicitly, and exhibit an approach to determining a minimal free resolution. Introduction What is the limit of the canonical model of a smooth curve as the curve degenerates to a hyperelliptic curve? "A ribbon" — more precisely "a ribbon on P1 " — may be defined as the answer to this riddle. A ribbon on P1 is a double structure on the projective line. Such ribbons represent a little-studied degeneration of smooth curves that shows promise especially for dealing with questions about the Clifford indices of curves. The theory of ribbons is in some respects remarkably close to that of smooth curves, but ribbons are much easier to construct and work with. In this paper we discuss the classification of ribbons and their maps. In particular, we construct the "holomorphic differentials" — sections of the canonical bundle — of a ribbon, and study properties of the canonical embedding. Aside from the genus, the main invariant of a ribbon is a number we call the "Clifford index", although the definition for it that we give is completely different from the definition for smooth curves. -
The Jouanolou-Thomason Homotopy Lemma
The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation. -
Connections on Bundles Md
Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. -
Ordinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license is that You are free to: Share copy and redistribute the material in any medium or format • Adapt remix, transform, and build upon the material for any purpose, even commercially. • The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes • were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Notices: You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation. No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. To view a full copy of this license, visit https://creativecommons.org/licenses/by/4.0/legalcode. -
Vector Bundles on Projective Space
Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. -
Canonical Bundle Formulae for a Family of Lc-Trivial Fibrations
CANONICAL BUNDLE FORMULAE FOR A FAMILY OF LC-TRIVIAL FIBRATIONS Abstract. We give a sufficient condition for divisorial and fiber space adjunc- tion to commute. We generalize the log canonical bundle formula of Fujino and Mori to the relative case. Contents 1. Introduction 1 2. Preliminaries 2 3. Lc-trivial fibrations 3 4. Proofs 6 References 6 1. Introduction A lc-trivial fibration consists of a (sub-)pair (X; B), and a contraction f : X −! Z, such that KZ + B ∼Q;Z 0, and (X; B) is (sub) lc over the generic point of Z. Lc-trivial fibrations appear naturally in higher dimensional algebraic geometry: the Minimal Model Program and the Abundance Conjecture predict that any log canonical pair (X; B), such that KX + B is pseudo-effective, has a birational model with a naturally defined lc-trivial fibration. Intuitively, one can think of (X; B) as being constructed from the base Z and a general fiber (Xz;Bz). This relation is made more precise by the canonical bundle formula ∗ KX + B ∼Q f (KZ + MZ + BZ ) a result first proven by Kodaira for minimal elliptic surfaces [11, 12], and then gen- eralized by the work of Ambro, Fujino-Mori and Kawamata [1, 2, 6, 10]. Here, BZ measures the singularities of the fibers, while MZ measures, at least conjecturally, the variation in moduli of the general fiber. It is then possible to relate the singu- larities of the total space with those of the base: for example, [5, Proposition 4.16] shows that (X; B) and (Z; MZ + BZ ) are in the same class of singularities. -
Foundations of Algebraic Geometry Class 46
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 46 RAVI VAKIL CONTENTS 1. Curves of genus 4 and 5 1 2. Curves of genus 1: the beginning 3 1. CURVES OF GENUS 4 AND 5 We begin with two exercises in general genus, and then go back to genus 4. 1.A. EXERCISE. Suppose C is a genus g curve. Show that if C is not hyperelliptic, then the canonical bundle gives a closed immersion C ,! Pg-1. (In the hyperelliptic case, we have already seen that the canonical bundle gives us a double cover of a rational normal curve.) Hint: follow the genus 3 case. Such a curve is called a canonical curve, and this closed immersion is called the canonical embedding of C. 1.B. EXERCISE. Suppose C is a curve of genus g > 1, over a field k that is not algebraically closed. Show that C has a closed point of degree at most 2g - 2 over the base field. (For comparison: if g = 1, it turns out that there is no such bound independent of k!) We next consider nonhyperelliptic curves C of genus 4. Note that deg K = 6 and h0(C; K) = 4, so the canonical map expresses C as a sextic curve in P3. We shall see that all such C are complete intersections of quadric surfaces and cubic surfaces, and con- versely all nonsingular complete intersections of quadrics and cubics are genus 4 non- hyperelliptic curves, canonically embedded. By Riemann-Roch, h0(C; K⊗2) = deg K⊗2 - g + 1 = 12 - 4 + 1 = 9: 0 P3 O ! 0 K⊗2 2 K 4+1 We have the restriction map H ( ; (2)) H (C; ), and dim Sym Γ(C; ) = 2 = 10. -
Recent Developments in Representation Theory
RECENT DEVELOPMENTS IN REPRESENTATION THEORY Wilfried 5chmid* Department of Mathematics Harvard University Cambridge, MA 02138 For the purposes of this lecture, "representation theory" means representation theory of Lie groups, and more specifically, of semisimple Lie groups. I am interpreting my assignment to give a survey rather loosely: while I shall touch upon various major advances in the subject, I am concentrating on a single development. Both order and emphasis of my presentation are motivated by expository considerations, and do not reflect my view of the relative importance of various topics. Initially G shall denote a locally compact topological group which is unimodular -- i.e., left and right Haar measure coincide -- and H C G a closed unimodular subgroup. The quotient space G/H then carries a G- invariant measure, so G acts unitarily on the Hilbert space L2(G/H). In essence, the fundamental problem of harmonic analysis is to decompose L2(G/H) into a direct "sum" of irreducibles. The quotation marks allude to the fact that the decomposition typically involves the continuous ana- logue of a sum, namely a direct integral, as happens already for non- compact Abelian groups. If G is of type I -- loosely speaking, if the unitary representations of G behave reasonably -- the abstract Plan- cherel theorem [12] asserts the existence of such a decomposition. This existence theorem raises as many questions as it answers: to make the decomposition useful, one wants to know it explicitly and, most impor- tantly, one wants to understand the structure of the irreducible sum- mands. In principle, any irreducible unitary representation of G can occur as a constituent of L2(G/H), for some H C G. -
Algebraic Topology
ALGEBRAIC TOPOLOGY C.R. F. MAUNDER ALGEBRAIC TOPOLOGY C. R. F. MAUNDER Fellow of Christ's College and University Lecturer in Pure Mathematics, Cambridge CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEW ROCHELLE MELBOURNE SYDNEY Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge C82inP 32East 57th Street, New York, NYzoozz,USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia CC. R. F. Miunder '970 CCambridge University Press 1980 First published by VanNostrandReinhold (UK) Ltd First published by the Cambridge University Press 1980 Firstprinted in Great Britain by Lewis Reprints Ltd, London and Tonbridge Reprinted in Great Britain at the University Press, Cambridge British Library cataloguing in publication data Maunder, Charles Richard Francis Algebraic topology. r.Algebraic topology I. Title 514'.2QA6!2 79—41610 ISBN 0521 231612 hard covers ISBN 0 521298407paperback INTRODUCTION Most of this book is based on lectures to third-year undergraduate and postgraduate students. It aims to provide a thorough grounding in the more elementary parts of algebraic topology, although these are treated wherever possible in an up-to-date way. The reader interested in pursuing the subject further will find ions for further reading in the notes at the end of each chapter. Chapter 1 is a survey of results in algebra and analytic topology that will be assumed known in the rest of the book. The knowledgeable reader is advised to read it, however, since in it a good deal of standard notation is set up. Chapter 2 deals with the topology of simplicial complexes, and Chapter 3 with the fundamental group. -
Formal GAGA for Good Moduli Spaces
FORMAL GAGA FOR GOOD MODULI SPACES ANTON GERASCHENKO AND DAVID ZUREICK-BROWN Abstract. We prove formal GAGA for good moduli space morphisms under an assumption of “enough vector bundles” (which holds for instance for quotient stacks). This supports the philoso- phy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms. 1. Introduction Good moduli space morphisms are a common generalization of good quotients by linearly re- ductive group schemes [GIT] and coarse moduli spaces of tame Artin stacks [AOV08, Definition 3.1]. Definition ([Alp13, Definition 4.1]). A quasi-compact and quasi-separated morphism of locally Noetherian algebraic stacks φ: X → Y is a good moduli space morphism if • (φ is Stein) the morphism OY → φ∗OX is an isomorphism, and • (φ is cohomologically affine) the functor φ∗ : QCoh(OX ) → QCoh(OY ) is exact. If φ: X → Y is such a morphism, then any morphism from X to an algebraic space factors through φ [Alp13, Theorem 6.6].1 In particular, if there exists a good moduli space morphism φ: X → X where X is an algebraic space, then X is determined up to unique isomorphism. In this case, X is said to be the good moduli space of X . If X = [U/G], this corresponds to X being a good quotient of U by G in the sense of [GIT] (e.g. for a linearly reductive G, [Spec R/G] → Spec RG is a good moduli space). In many respects, good moduli space morphisms behave like proper morphisms. They are uni- versally closed [Alp13, Theorem 4.16(ii)] and weakly separated [ASvdW10, Proposition 2.17], but since points of X can have non-proper stabilizer groups, good moduli space morphisms are gen- erally not separated (e.g. -
General Introduction to K3 Surfaces
General introduction to K3 surfaces Svetlana Makarova MIT Mathematics Contents 1 Algebraic K3 surfaces 1 1.1 Definition of K3 surfaces . .1 1.2 Classical invariants . .3 2 Complex K3 surfaces 9 2.1 Complex K3 surfaces . .9 2.2 Hodge structures . 11 2.3 Period map . 12 References 16 1 Algebraic K3 surfaces 1.1 Definition of K3 surfaces Let K be an arbitrary field. Here, a variety over K will mean a separated, geometrically integral scheme of finite type over K.A surface is a variety of dimension two. If X is a variety over of dimension n, then ! will denote its canonical class, that is ! =∼ Ωn . K X X X=K For a sheaf F on a scheme X, I will write H• (F) for H• (X; F), unless that leads to ambiguity. Definition 1.1.1. A K3 surface over K is a complete non-singular surface X such that ∼ 1 !X = OX and H (X; OX ) = 0. Corollary 1.1.1. One can observe several simple facts for a K3 surface: ∼ 1.Ω X = TX ; 2 ∼ 0 2.H (OX ) = H (OX ); 3. χ(OX ) = 2 dim Γ(OX ) = 2. 1 Fact 1.1.2. Any smooth complete surface over an algebraically closed field is projective. This fact is an immediate corollary of the Zariski{Goodman theorem which states that for any open affine U in a smooth complete surface X (over an algebraically closed field), the closed subset X nU is connected and of pure codimension one in X, and moreover supports an ample effective divisor.