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Real Algebraic Varieties

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This book gives a systematic presentation of real algebraic varieties.

Real algebraic varieties are ubiquitous.They are the first objects encountered when learning of coordinates, then equations, but the systematic study of these objects, however elementary they may be, is formidable.

This book is intended for two kinds of audiences: it accompanies the reader, familiar with algebra and geometry at the masters level, in learning the basics of this rich theory, as much as it brings to the most advanced reader many fundamental results often missing from the available literature, the “folklore”. In particular, the introduction of topological methods of the theory to non-specialists is one of the original features of the book.

The first three chapters introduce the basis and classical methods of real and complex algebraic geometry. The last three chapters each focus on one more specific aspect of real algebraic varieties. A panorama of classical knowledge is presented, as well as major developments of the last twenty years in the topology and geometry of varieties of dimension two and three, without forgetting curves, the central subject of Hilbert's famous sixteenth problem.

Various levels of exercises are given, and the solutions of many of them are provided at the end of each chapter.

Author(s): Frédéric Mangolte
Series: Springer Monographs in Mathematics
Publisher: Springer
Year: 2020

Language: English
Pages: 464
City: Cham

Preface
Introduction
Algebraic Models of Smooth Manifolds
Nash and Tognoli Theorems
Contents
List of Figures
1 Algebraic Varieties
1.1 Algebraic Varieties: Points or Spectra?
1.2 Affine and Projective Algebraic Sets
1.3 Abstract Algebraic Varieties
1.4 Euclidean Topology
1.5 Dimension and Smooth Points
1.6 Plane Curves
1.7 Umbrellas
1.8 Solutions to exercises of Chapter 1
2 mathbbR-Varieties
2.1 Real Structures on Complex Varieties
2.2 mathbbR-Varieties and Real Algebraic Varieties
2.2.1 Non Singular mathbbR-Varieties
2.2.2 Compatible Atlas
2.3 Complexification of a Real Variety
2.3.1 Rational Varieties
2.4 mathbbR-Varieties, Real Algebraic Varieties and Schemes Over mathbbR—a Comparison
2.4.1 Real Forms of a mathbbC-Scheme
2.4.2 Notations X, X(mathbbR), X(mathbbC), XmathbbC, XmathbbR
2.5 Coherent Sheaves and Algebraic Bundles
2.5.1 Coherent mathbbR-Sheaves
2.5.2 Algebraic Vector Bundles
2.6 Divisors on a Projective mathbbR-Variety
2.6.1 Weil Divisors
2.6.2 Cartier Divisors
2.6.3 Line Bundles
2.6.4 Galois Group Action on the Picard Group
2.6.5 Projective Embeddings
2.6.6 Review of Theorem 2.1.33
2.6.7 Nakai–Moishezon Criterion
2.6.8 Degree of a Subvariety of Projective Space
2.7 mathbbR-Plane Curves
2.8 Solutions to exercises of Chapter 2
3 Topology of Varieties with an Involution
3.1 Homology and Cohomology of mathbbR-Varieties
3.1.1 Involutive Modules
3.1.2 Poincaré Duality on mathbbR-Varieties
3.1.3 Orientability and Characteristic Classes
3.2 Smith Theory
3.3 Upper Bounds on Betti Numbers
3.3.1 Singular Curves
3.4 The Intersection Form on an Even-Dimensional mathbbR-Variety
3.4.1 Surfaces in mathbbP3
3.5 Classification of mathbbR-Curves and xvith Hilbert's Problem
3.5.1 Abstract Classification
3.5.2 First Part of xvith Hilbert's Problem
3.5.3 Ragsdale's Conjecture
3.5.4 Construction of a Maximal Quartic mathbbR-surface
3.6 Galois-Maximal Varieties
3.6.1 σ-Representable Classes and Proof of Theorem 3.6.11
3.7 Algebraic Cycles
3.7.1 Fundamental Class
3.7.2 Algebraic Cycles
3.7.3 Cycle Map
3.7.4 Applications to mathbbR-Surfaces
3.8 Solutions to exercises of Chapter3
4 Surfaces
4.1 Curves and Divisors on Complex Surfaces
4.1.1 Intersection Form
4.1.2 Blow-Up
4.1.3 Adjunction Formula
4.1.4 Genus of an Embedded Curve
4.2 Examples of mathbbR-Surfaces
4.2.1 Topological Surfaces: Conventions and Notations
4.3 mathbbR-Minimal Surfaces
4.3.1 Deformation Families
4.4 Uniruled and Rational Surfaces (κ=-infty)
4.4.1 Rational mathbbR-Surfaces
4.4.2 Singular Surfaces and Parabolas
4.4.3 Generalisation of Comessatti's Theorem
4.5 K3, Enriques, Abelian and Bi-elliptic Surfaces (κ=0)
4.5.1 K3 Surfaces
4.5.2 Algebraic Cycles on K3 Surfaces
4.5.3 Enriques Surfaces
4.5.4 Algebraic Cycles on Enriques Surfaces
4.5.5 Abelian Surfaces
4.5.6 Algebraic Cycles on Abelian Surfaces
4.5.7 Bi-elliptic Surfaces
4.5.8 Algebraic Cycles on Bi-elliptic Surfaces
4.5.9 Summary: Algebraic Cycles on Surfaces with κleqslant0
4.6 Elliptic Surfaces (κleqslant1)
4.6.1 Algebraic Cycles on Elliptic Surfaces such that q=0
4.7 Surfaces of General Type (κ=2)
4.7.1 Resolution of Singular Points and Double Covers
4.7.2 Resolutions of Real Double Covers
4.8 Solution to exercises of Chapter 4
5 Algebraic Approximation
5.1 Rational Models
5.2 Smooth and Regular Maps
5.2.1 Homotopy, Approximations and Algebraic Bundles
5.3 Maps to Spheres
5.3.1 Maps to mathbbS1
5.3.2 Maps to mathbbS2
5.3.3 Regulous Functions
5.4 Diffeomorphisms and Biregular Maps
5.4.1 Rational Models
5.4.2 Automorphisms of the Real Locus
5.4.3 Cremona Groups of Real Surfaces
5.4.4 Density of Birational Diffeomorphisms
5.4.5 Approximation by Rational Curves
5.5 Fake Real Planes
6 Three Dimensional Varieties
6.1 The Nash Conjecture from 1952 to 2000 via 1914
6.1.1 Rational Varieties
6.1.2 The Nash Conjecture for Surfaces
6.1.3 The Topological Nash Conjecture Holds
6.1.4 The Projective Singular Nash Conjecture Holds if nleqslant3
6.1.5 Non Projective Non Singular Nash Holds for n=3
6.1.6 Non Singular Projective Nash Fails for All Dimensions n>1
6.2 Real Uniruled 3-varieties from 2000 to 2012
6.2.1 Uniruled Varieties
6.2.2 Rationally Connected Varieties
6.3 Questions and Conjectures
Appendix A Commutative Algebra
A.1 Inductive Limits
A.2 Rings, Prime Ideals, Maximal Ideals and Modules
A.3 Localisation
A.4 Tensor Product
A.5 Rings of Integers and the Nullstellensatz
A.5.1 The Nullstellensatz over an Algebraically Closed Field
A.5.2 The Nullstellensatz over a Real Closed Field
A.6 Quadratic mathbbZ-Modules and Lattices
A.7 Anti-linear Involutions
A.8 Solution to Exercises of Appendix A
Appendix B Topology
B.1 Hausdorff Spaces
B.2 Semi-algebraic Sets
B.3 Simplicial Complexes and Homology
B.4 Universal Coefficients Theorem
B.5 Topological and Differentiable Manifolds and Orientability
B.5.1 Connected Sums
B.5.2 Spin Structures
B.5.3 Topologies on a Family of Maps
B.6 Cohomology
B.6.1 Cohomology with Compact Support
B.7 Poincaré Duality
B.7.1 Application: Orientability of a Submanifold
B.7.2 Applications to Algebraic Varieties
B.7.3 Compact ANRs
B.8 Three Dimensional Manifolds
B.8.1 Seifert Manifolds
B.8.2 Lens Spaces
B.8.3 mathcalCinfty Geometric Manifolds
B.8.4 Geometrisation and Classification
Classification of mathcalCinfty Manifolds of Dimension 3
Appendix C Sheaves and Ringed Spaces
C.1 Sheaves
C.2 Sheaf Spaces over X
C.3 Stalks of a Sheaf
C.3.1 Locally Trivial Fibrations
C.3.2 Sheaf Morphisms
C.4 Sheaf of Sections of a Sheaf Space
C.5 Ringed Spaces
C.6 Coherent Sheaves
C.7 Algebraic Varieties over an Algebraically Closed Base Field
Appendix D Analytic Geometry
D.1 Complex Analytic Spaces and Holomorphic Functions
D.2 Complex Analytic Varieties
D.2.1 Stein Manifolds
D.2.2 Serre Duality
D.3 Kähler Manifolds and Hodge Theory
D.3.1 Hodge Theory
D.3.2 De Rham's Theorem
D.3.3 Dolbeault's Theorem
D.3.4 Hodge Decomposition
D.3.5 Consequences
D.4 Numerical Invariants
D.5 Projective Varieties
D.6 Picard and Albanese Varieties
D.6.1 Picard Variety
D.6.2 Albanese Variety
D.7 Riemann–Roch Theorem
D.7.1 Riemann–Roch for Curves
D.7.2 Riemann–Roch on Surfaces
D.8 Vanishing Theorems
D.9 Other Fundamental Theorems
Appendix E Riemann Surfaces and Algebraic Curves
E.1 Genus and Topological Classification of Surfaces
E.2 Complex Curves and Riemann Surfaces
E.3 The Riemann–Roch Theorem for a Curve
E.4 Jacobian Variety Associated to a Curve
Appendix F Blow Ups
F.1 Blowing Up mathcalCinfty Manifolds
F.1.1 Tautological Bundle
F.1.2 Projectivisation of the Normal Bundle
F.1.3 Blowing Up a Manifold Along a Submanifold
F.2 Blow Ups of Algebraic Varieties
F.2.1 Strict Transform
F.3 Topology of Blow Ups
Glossary of Notations
References
Index
List of Examples