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黎曼曲面和热带曲线的模空间导引(英文版)

样章
作者：
Lizhen Ji,Eduard Looijenga
定价：
69.00元
ISBN：
978-7-04-047419-0
版面字数：
240千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2017-04-28
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数几何学

黎曼曲面及其模空间的概念由黎曼分别在其博士毕业论文和一篇著名的文章中定义。由于同数学与物理的许多学科联系广泛，黎曼曲面及其模空间得到了深入的研究，并将继续吸引人们的关注。近期热带曲线的研究迅速崛起。热带代数曲线是经典复数域上代数曲线以及黎曼曲面在热带半环上的一种模拟。

本书深入浅出地介绍了以上几个重要学科，并且重点强调如代数几何、复几何、双曲几何、拓扑、几何群理论和数学物理等不同学科之间的关联。

  • preface
  • Part I Moduli Spaces of Riemann Surfaces
    • 1 Mapping class groups and Dehn twists
      • Mapping class groups
      • Dehn twists
      • Fundamental groups and mapping class groups
    • 2 Conformal structures and a rough classification
      • Conformal structures
      • Spherical cases
      • Flat cases
      • Hyperbolic cases
    • 3 Geometry of the upper half plane
    • 4 Hyperbolic surfaces
      • Closed geodesics on a hyperbolic surface
      • Geodesic shear
      • Geodesic representation
      • Disjunction of geodesics
      • Pair of pants
      • Pants decompositions
      • Fenchel-Nielsen coordinates
      • Hyperbolic surfaces with cusps
      • Pants decomposition in the presence of cusps
    • 5 Quadratic differentials
      • Local normal form
      • Structure defined by a holomorphic quadratic differential
      • Quadratic differentials and the Teichm¨uller flow
    • 6 Ribbon graphs and ideal triangulation of Teichm¨uller space
      • Jenkins-Strebel differentials
      • Combinatorial description of a graph
      • Ribbon graphs
      • The dual ribbon graph and the arc complex
      • Metrized ribbon graphs
      • Ideal triangulation of thickened Teichm¨uller space
    • 7 The homotopy type of the moduli spacesMg,n
      • The homotopy type ofMg,1
      • The moduli space of curves as a virtual classifying space
    • 8 A brief review of Dolbault cohomology and Serre duality for Riemann surfaces
      • Dolbault cohomology of a coherent sheaf
      • Riemann-Roch and Serre duality for a Riemann surface
    • 9 Deformation theory of Riemann surfaces
      • Orbifolds
      • Towards the complex tangent bundle of Teichm¨uller space
      • Deformation theory
      • The Kodaira Spencermap
      • TheWeil-Petersson metric
      • The quasi-projective structure on the universal family
    • 10 Harvey bordification and Deligne-Mumford compactification
      • The curve complex
      • Harvey’s bordification
      • Stable pointed curves
      • Deformations of nodes
      • Deformations of nodal curves
      • The Deligne-Mumford compactification
      • The idea behindGeometric Invariant Theory
      • Projectivity of the moduli space of stable pointed curves
      • Getting Harvey’s bordification from the D-M compactification
    • 11 Cohomological properties ofMg,P
      • Harer’s stability theorem
      • Hopf algebra structure on stable cohomology
    • 12 Tautological algebras
      • Duality on orbifolds
      • Review of the Gysinmap
      • The moduli space of stable pointed curves as a category
      • Vector bundles on moduli spaces of curves
      • Weighted graphs
      • The notion of a tautological algebra
      • A theoremof Kontsevich
      • Faber’s conjectures
      • Primitivity of the kappa classes
    • References
  • Part II Introduction to Tropical Curves, Tropical Moduli and Teichm¨ uller Spaces
    • 1 Introduction
      • 1.1 A general overview
      • 1.2 A specific motivation fromgeometric group theory
      • 1.3 Interaction between moduli spaces of tropical curves and outer space ofmetric graphs
    • 2 Definitions and basic facts on Out(Fn) and the outer space Xn
      • 2.1 Combinatorial group theory and motivations from mapping class groups
      • 2.2 Geometric group theory, metric graphs and outer space Xn
      • 2.3 Action of Out(Fn) on outer space Xn and applications
      • 2.4 A natural approach to construct complete geodesic metrics on Xn
    • 3 Tropical semifields and tropical polynomials
      • 3.1 The tropical semifield T
      • 3.2 Tropical polynomials
    • 4 Tropical curves
      • 4.1 Plane tropical curves Γp
      • 4.2 Newton polygons and tropical curves
      • 4.3 Balancing condition at vertices
      • 4.4 Abstract smooth tropical curves
      • 4.5 Identification between smooth tropical curves and metric graphs
    • 5 Definitions of tropical varieties andmotivations
      • 5.1 Tropical hypersurfaces and higher codimension tropical subvarieties of Rn
      • 5.2 Tropicalization of polynomials and tropical varieties as non-Archimedean amoebas
      • 5.3 Tropical varieties as logarithmic limit sets
      • 5.4 Tropical varieties as limits of Archimedean amoebas
      • 5.5 Affine manifolds and tropical affinemanifolds
    • 6 Some alternative descriptions of tropical varieties and related spaces
      • 6.1 Fans and tropical fans
      • 6.2 Stacky fans
      • 6.3 Tropical hypersurfaces as super currents
      • 6.4 Tropical varieties versus Berkovich spaces
    • 7 Applications of tropical geometry
      • 7.1 Logarithmic compactification of algebraic varieties
      • 7.2 Topology of real algebraic curves
      • 7.3 Geometric and combinatorial group theory
      • 7.4 Three-dimensional topology
      • 7.5 Boundary of Teichm¨uller space Tg
      • 7.6 Enumerative algebraic geometry
      • 7.7 Mirror symmetry of Calabi-Yaumanifolds
      • 7.8 Arithmetic algebraic geometry
    • 8 Moduli spaces of tropical curves and tropical Teichm¨uller spaces
      • 8.1 Moduli space Mtr
      • n of tropical curves and a general philosophy on moduli spaces
      • 8.2 Gromov-Hausdorff distance and topology onMtr n
      • 8.3 The moduli spaceMtr
      • n and tropical Teichm¨uller space T tr g as stacky fans
      • 8.4 The tropical Teichm¨uller space T tr g as a tropical space
      • 8.5 The moduli spaceMtr n as a tropical orbifold
      • 8.6 Compactifications of the moduli spaceMtr n
    • 9 Jacobian variety of a Riemann surface and the Siegel upper-half space
      • 9.1 Polarized algebraic varieties
      • 9.2 Abelian varieties
      • 9.3 Jacobian varieties of Riemann surfaces
      • 9.4 Moduli space of principally polarized abelian varieties Ag
    • 10 Tropical abelian varieties, their moduli spaces, and Jacobians of tropical curves
      • 10.1 Jacobian variety of tropical curves andmetric graphs
      • 10.2 Tropical abelian varieties
      • 10.3 Principally polarized tropical abelian varieties and their moduli space Atr n
    • 11 The Torelli Theorem for tropical curves
      • 11.1 Period of tropical curves and contractions of graphs
      • 11.2 Tropical Torelli theoremand its failure
    • 12 Complete invariant metrics on outer space Xn
      • 12.1 Tropical Jacobian map and invariant complete geodesic metrics on Xn
      • 12.2 Tropical Jacobian map and a complete pesudo-Riemannian metric on Xn
      • 12.3 Complete pseudo-Riemannian metric on Xn via lengths of pinching loops
      • 12.4 Finite Riemannian volume of the quotient Out(Fn)\Xn
    • References
  • Index

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