نقاط گ rational، منحنی‌های rational و منحنی‌های هولومورفیک تمام‌نما بر روی گونه‌های پروژه‌ای: برنامهٔ موضوعی کوتاه CRM دربارهٔ نقاط گ rational، منحنی‌های rational و منحنی‌های هولومورفیک تمام‌نما و گونه‌های جبری، ۳-۲۸ ژوئن ۲۰۱۳، مرکز تحقیقات ریاضی

This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3–28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. Cover 1 Title page 4 Contents 6 Preface 8 Expository and survey articles 10 Some applications of p-adic uniformization to algebraic dynamics 12 1. A motivation: potential density 12 2. Near a fixed point 16 3. Uniformization of orbits and applications 21 4. Acknowledgments 28 References 29 Special manifolds, arithmetic and hyperbolic aspects: a short survey 32 1. Introduction 32 2. Special manifolds: Bogomolov sheaves 33 3. Special manifolds: orbifold base 37 4. The orbifold version C^{orb}_{n,m} of the C_{n,m} conjecture 40 5. The core fibration 41 6. The decomposition c=(Jr)n of the core 43 7. Conjectures for smooth and integral orbifolds 49 8. Examples of fibrations of general type on simply-connected manifolds 54 9. Orbifold cotangent sheaves: generic semi-positivity, applications 55 10. H-principle and specialness 58 References 59 Invitation to integral and rational points on curves and surfaces 62 1. Introduction 62 2. Some Diophantine equations 63 3. Diophantine geometry - rational and integral points on curves 67 4. Rational and integral points on surfaces 75 References 79 Roth’s theorem: an introduction to diophantine approximation 84 0. Introduction 84 1. Liouville’s theorem and beyond 86 2. Roth’s theorem: an overview 93 3. Controlling the index: Roth’s lemma 98 4. Controlling the index: Dyson’s lemma 103 5. Extensions and questions 109 References 115 The Thue-Siegel method in diophantine geometry 118 1. The Thue-Siegel method 120 2. Basic constructions in Bombieri’s proof 125 3. Divisors, heights, and Jacobians 126 4. An upper bound for the height 128 5. A lower bound 129 6. Construction of a global section 132 7. The index 134 8. The end of the proof 135 References 137 Research articles 140 Optimal pinching for the holomorphic sectional curvature of Hitchin’s metrics on Hirzebruch surfaces 142 1. Introduction 142 2. Basic definitions and description of the family of metrics under consideration 143 3. Proof of Theorem 1.1 145 4. Geometric interpretation of our computations 149 5. Proof of Theorem 1.2 149 References 151 The Lefschetz property for families of curves 152 References 162 Separable rational connectedness and stability 164 References 168 Curve classes on rationally connected varieties 170 1. Introduction 170 2. Preliminaries 171 3. Proof of the Main Theorem 172 References 174 Back Cover 176 This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. Read more The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation