Lectures in Real Geometry

Description

Foreword Introduction Basic algorithms in real algebraic geometry and their complexity: from Sturm’s theorem to the existential theory of reals 1. Introduction 2. Real closed fields 2.1. Definition and first examples of real closed fields 2.2. Cauchy index and real root counting 3. Real root counting 3.1. Sylvester sequence 3.2. Subresultants and remainders 3.3. Sylvester-Habicht sequence 3.4. Quadratic forms, Hankel matrices and real roots 3.5. Summary and discussion 4. Complexity of algorithms 5. Sign determinations 5.1. Simultaneous inequalities 5.2. Thom’s lemma and its consequences 6. Existential theory of reals 6.1. Solving multivariate polynomial systems 6.2. Some real algebraic geometry 6.3. Finding points on hypersurfaces 6.4. Finding non empty sign conditions References Nash functions and manifolds §1. Introduction §2. Nash functions §3. Approximation Theorem §4. Nash manifolds §5. Sheaf theory of Nash function germs §6. Nash groups References Approximation theorems in real analytic and algebraic geometry Introduction I. The analytic case 1. The Whitney topology for sections of a sheaf 2. A Whitney approximation theorem 3. Approximation for sections of a sheaf 4. Approximation for sheaf homomorphisms II. The algebraic case 5. Preliminaries on real algebraic varieties 6. A- and B-coherent sheaves 7. The approximation theorems in the algebraic case III. Algebraic and analytic bundles 8. Duality theory 9. Strongly algebraic vector bundles 10. Approximation for sections of vector bundles References Real abelian varieties and real algebraic curves Introduction 1. Generalities on complex tori 1.1. Complex tori 1.2. Homology and cohomology of tori 1.3. Morphisms of complex tori 1.4. The Albanese and the Picard variety 1.5. Line bundles on complex tori 1.6. Polarizations 1.7. Riemann’s bilinear relations and moduli spaces 2. Real structures 2.1. Definition of real structures 2.2. Real models 2.3. The action of conjugation on functions and forms 2.4. The action of conjugation on cohomology 2.5. A theorem of Comessatti 2.6. Group cohomology 2.7. The action of conjugation on the Albanese variety and the Picard group 2.8. Period matrices in pseudonormal form and the Albanese map 3. Real abelian varieties 3.1. Real structures on complex tori 3.2. Equivalence classes for real structures on complex tori 3.3. Line bundles on complex tori with a real structure 3.4. Riemann bilinear relations for principally polarized real varieties 3.5. Moduli spaces of principally polarized real abelian varieties 3.6. Real theta functions 4. Applications to real curves 4.1. The Jacobian of a real curve 4.2. Real theta-characteristics 4.3. Examples 4.4. Moduli spaces and the theorem of Torelli 4.5. Singular curves References Appendix Mario Raimondo’s contributions to real geometry Mario Raimondo’s contributions to computer algebra