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
