Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben

2013 | Buch

Schemes Kapitel lesen Erstes Kapitel lesen

Basic Algebraic Geometry 2

Schemes and Complex Manifolds

verfasst von: Igor R. Shafarevich

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.''

The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the ``Shafarevich conjecture''.

The style of Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.

Inhaltsverzeichnis

Frontmatter

Book 2: Schemes and Varieties

Frontmatter
Chapter 5. Schemes
Abstract
The chapter introduces schemes as the new foundational object in algebraic geometry, with an extensive discussion of the ideas underlying this new notion. The prime spectrum SpecA of an arbitrary commutative ring with a 1 is defined as the set of prime ideals of A. It has a Zariski topology and a structure sheaf, a sheaf of rings with stalk at a point \({\frak{p}}\) the local ring \(A_{{\frak{p}}}\). Several examples are discussed, along with foundation notions, such as the dimension and product of affine schemes. General schemes are defined, together with the notion of product of schemes and the separatedness axiom in terms of closure of the diagonal.
Igor R. Shafarevich
Chapter 6. Varieties
Abstract
Scheme theory provides the modern definition of a variety, and a convenient language for all the constructions of algebraic geometry, ancient and modern. A variety over an algebraically closed field k is a separated reduced scheme of finite type over k. The general properties of quasiprojective varieties from Volume 1 of the book are reinterpreted in this intrinsic framework.
There follows a comparison between varieties and projective varieties, including an intrinsic treatment of blowups, Chow’s lemma that any variety has a blowup that is quasiprojective, and a brief discussion of different criteria for a variety to be projective. On the other hand, an example is given (and illustrated on the front cover of Volume 2) of a complete variety that cannot be embedded in any projective space.
The chapter also discusses in some detail two other circles of ideas: sheaves of modules, including locally free sheaves and coherent sheaves, and the idea of a scheme representing a functor, that plays an central role in the modern theory of moduli.
Igor R. Shafarevich

Book 3: Complex Algebraic Varieties and Complex Manifolds

Frontmatter
Chapter 7. The Topology of Algebraic Varieties
Abstract
A variety over \(\mathbb{C}\) has a Euclidean topology defining an underlying topological space which, for a nonsingular variety, is a real differentiable manifold, that is orientable and of twice the complex dimension. The variety inherits the usual topological invariants such as fundamental group and cohomology. In this context, complete implies compact. A set of less elementary questions centres around the connectedness of the topological space underlying an irreducible variety. The chapter includes a sketch of the local and relative versions of this problem, that includes the Zariski connectedness theorem.
The topology of algebraic curves leads to the famous picture of a compact Riemann surfaces as a sphere with g handles and Euler characteristic 2−2g. The chapter also discusses the geometry of the nested ovals of real algebraic plane curves and the possible complex conjugation maps.
Igor R. Shafarevich
Chapter 8. Complex Manifolds
Abstract
A variety over \(\mathbb{C}\) in the Euclidean topology also has an analytic structure sheaf consisting of local holomorphic functions that makes it into a complex space, or a complex manifold if the variety is nonsingular. Many features of the geometry of algebraic varieties carry over to the complex analytic setting, such as the link between divisors and line bundles. The relation is especially close for complex manifolds that arise from complete varieties. Then global meromorphic functions are rational functions, analytically defined divisors and sheaves are algebraic, and two varieties are isomorphic as algebraic varieties if and only if the corresponding complex manifolds are isomorphic. However, in dimension ≥2 there are many complex manifolds not arising from algebraic varieties, for example, some having no nonconstant global meromorphic functions or no nontrivial submanifolds; there are also algebraic varieties (necessarily noncomplete) whose complex spaces are isomorphic, but that are not isomorphic as varieties. The first three sections discuss these ideas, with examples constructed as quotient manifolds by a discrete group action.
The final section contains a discussion from first principles of Kähler metrics on complex manifolds. A Kähler metric on a complex manifold is a natural additional structure that makes it closer to a projective algebraic variety. The chapter concludes with a description of the Hodge structure on the cohomology of a Kähler manifold.
Igor R. Shafarevich
Chapter 9. Uniformisation
Abstract
The chapter discusses what is known about the fundamental group and universal cover of compact complex manifold. For algebraic curves, the primary theory is classical: a curve of genus 0 is isomorphic to \(\mathbb{P}^{1}\), by the Riemann mapping theorem, curves of genus 1 are uniformised by \(\mathbb{C}\) with the fundamental group a lattice of translations, and curves of genus ≥2 by the upper half-plane, with the covering group a cocompact discrete subgroup of \(\mathop{{\mathrm{SL}}}(2,\mathbb{R})\). Conversely, given a cocompact discrete group acting on any bounded domain (of any dimension), the quotient is a projective algebraic variety, and has pluricanonical embeddings into projective space provided by Poincaré series.
In higher dimensions the theory is much more fragmentary. Standard constructions of projective geometry such as complete intersections lead to simply connected varieties. By taking appropriate group quotients of these, one can obtain every finite group as the fundamental group of a compact complex manifold. The final section raises the question (now considered to be a deep and studied under the name of Shafarevich’s conjecture) of whether the universal cover of a complete algebraic variety is holomorphically convex.
Igor R. Shafarevich
Backmatter
Metadaten
Titel
Basic Algebraic Geometry 2
verfasst von
Igor R. Shafarevich
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-38010-5
Print ISBN
978-3-642-38009-9
DOI
https://doi.org/10.1007/978-3-642-38010-5

Premium Partner

    Bildnachweise