由季理真、路易安嘎著的《黎曼曲面和熱帶曲線
的模空間導引(英文版)(精)》黎曼曲面及其模空間的
概念由黎曼分別在其博士畢業論文和一篇著名的文章
中定義。由於同數學與物理的許多學科聯繫廣泛，黎
曼曲面及其模空間得到了深入的研究，並將繼續吸引
人們的關注。近期熱帶曲線的研究迅速崛起。熱帶代
數曲線是經典復數域上代數曲線以及黎曼曲面在熱帶
半環上的一種模擬。本書深入淺出地介紹了以上幾個
重要學科，並且重點強調如代數幾何、復幾何、雙曲
幾何、拓撲、幾何群理論和數學物理等不同學科之間
的關聯。

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 Teichmtiller flow
6 Ribbon graphs and ideal triangulation of Teichmuller 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 Teichmtuler space
7 The homotopy type of the moduli spaces □
The homotopy type of □
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 Teichmiiller space
Deformation theory
The Kodaira Spencer map
The Weil-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 behind Geometric Invariant Theory
Projectivity of the moduli space of stable pointed curves
Getting Harvey's bordification from the D-M compactification...
11 Cohomological properties of □
Harer's stability theorem
Hopf algebra structure on stable cohomology
12 Tautological algebras
Duality on orbifolds
Review of the Gysin map
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 theorem of Kontsevich
Faber's conjectures
Primitivity of the kappa classes
References
Part II Introduction to Tropical Curves, Tropical Moduli and
Teichmiiller Spaces
1 Introduction
1.1 A general overview
1.2 A specific motivation from geometric group theory
1.3 Interaction between moduli spaces of tropical curves and
outer space of metric 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 гp
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 and motivations
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 affine manifolds
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 Teichmuller space □
7.6 Enumerative algebraic geometry
7.7 Mirror symmetry of Calabi-Yau manifolds
7.8 Arithmetic algebraic geometry
8 Moduli spaces of tropical curves and tropical Teichmuller spaces...
8.1 Moduli space □ of tropical curves and a general
philosophy on moduli spaces
8.2 Gromov-Hausdorffdistance and topology on □
8.3 The moduli space □ and tropical Teichmuller space □
as stacky fans
8.4 The tropical Teichmiiller space □ as a tropical space
8.5 The moduli space □ as a tropical orbifold
8.6 Compactifications of the moduli space □
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 □
10 Tropical abelian varieties, their moduli spaces, and Jacobians of
tropical curves
10.1 Jacobian variety of tropical curves and metric graphs
10.2 Tropical abelian varieties
10.3 Principally polarized tropical abelian varieties and their
moduli space □
11 The Torelli Theorem for tropical curves
11.1 Period of tropical curves and contractions of graphs
11.2 Tropical Torelli theorem and 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