Summer school on algebraic geometry, held at santa cruz in july 1995. A complex manifold x is called kobayashi hyperbolic if its kobayashi pseudodistance is nondegenerate. This is analogous to the the half space model of h2 r. Complex hyperbolic geometry is a particularly rich field, drawing on riemannian geometry, complex analysis, symplectic and contact geometry, lie group theory, and harmonic analysis. For statements involving the vanishing of, its vanishing in the second interpretation for all choices of the coordinate. The boundary in complex hyperbolic spaces, known as spherical cr or heisenberg geometry, reflects this richness. Another standard model for complex hyperbolic space is a paraboloid in c2 called the siegel domain.
Request pdf hyperbolicity in complex geometry a complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. Most of the time we will use the latter interpretation. We discuss the techniques and methods for the hyperbolicity problems for submanifolds and their complements in abelian varieties and the complex projective space. Mathematics, math research, mathematical modeling, mathematical programming, math articles, applied math. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of tr. Aspects in complex hyperbolicity purdue university. Chapter 1 geometry of real and complex hyperbolic space 1. Complex structures in algebra, geometry, topology, analysis. International conference in nevanlinna theory and complex geometry march 14 18, 2012. Hyperbolicity is a largescale property, and is very useful to the study of certain. We discuss the techniques and methods for the hyperbolicity problems for submanifolds and their complements in abelian varieties and the complex projective. On the hyperbolicity of surfaces of general type with small k.
Weilpetersson metric and hyperbolicity problems of some families of polarized manifolds conference on complex geometry, dynamic systems and foliation theory institute for mathematical sciceces national university of singapore may 1519, 2017 saikee yeung purdue university may 16, 2017. Classical hyperbolic spacecat0 spacescube complexesadvantages of cat0 geometry importance in group theory the group of isometries of the the poincar e disk is the lie group psl 2r, so studying hyperbolic geometry can give us information about this group and other related lie groups. Hyperbolic geometry was created in the rst half of the nineteenth century in the midst of attempts to understand euclids axiomatic basis for geometry. The two commonly used notions are brody hyperbolicity and kobayashi hyperbolicity. Instead, we will develop hyperbolic geometry in a way that emphasises the similarities and more interestingly. Dirac geometry is based on the idea of unifying the geometry of a poisson structure with that of a closed 2form, whereas generalized complex geometry unifies complex and symplectic geometry. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number. Schedule revised 3812 all talks held in 127 hayeshealy hall. We show how the global clustering coe cient can be. Hyperbolicity in complex geometry 3 on c which is the coe. Kobayashi hyperbolicity implies brody hyperbolicity. Einstein and minkowski found in noneuclidean geometry a.
Complex hyperbolic geometry oxford mathematical monographs. We develop a geometric framework to study the structure and function of complex networks. A similar relative hyperbolicity result is proved for the mapping class group of a surface. It was introduced by harvey as an analogy, in the context of. On hyperbolic geometry structure of complex networks.
These lecture notes are based on a minicourse given. X y be a smooth morphism of complex manifolds, where. They have profound applications to the study of complex variables, to the topology of two and threedimensional manifolds, to the study of nitely presented in nite groups, to physics, and to other disparate elds of mathematics. Everything from geodesics to gaussbonnet, starting with a. Incident, parallel, and ultraparallel subspaces 39 2. The notion of a hyperbolic metric space was introduced by gromov gr1. Then holomorphic discs in xsatisfy an isoperimetric linear inequality, i. The hyperbolicity problem in complex geometry studies the conditions for a given complex manifold x to be hyperbolic.
However, while there are a number of books on analysis in such spaces, this. Free geometry books download ebooks online textbooks. In hyperbolic geometry, the circumference of a circle of radius r is greater than. Hyperbolicity is a quasiisometry invariant, from which one can deduce immediately that certain variations on the curve complex are also hyperbolic 1. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still. Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. We do a rigorous analysis of clustering and characterize the global clustering coe cient in terms of the parameters of the model. For compact complex manifolds the converse of this result is true br2. Introduction consider a compact oriented surface s of genus g. Thurston introduced tools from hyperbolic geometry to study knots that led to new geometric invariants, especially hyperbolic volume. Clustering and the hyperbolic geometry of complex networks. Basics of hyperbolic geometry rich schwartz october 8, 2007 the purpose of this handout is to explain some of the basics of hyperbolic geometry.
Pdf hypercomplex hyperbolic geometry sarah markham. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. The complex of curves exactly encodes the intersection patterns of this family of regions it is the nerve of the family, and we show that its hyperbolicity means that the teichmuller space is relatively hyperbolic with respect to this family. Further explanation of complex networks and hyperbolicity will be given in the rest of this section. Everyone who takes a course in complex analysis learns the schwarz lemma. The geometry of knot complements city university of new york. Thoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity includes full solutions for all exercises successful first edition sold over 800 copies in north america. Hyperbolicity in complex geometry harvard mathematics. In mathematics and especially complex geometry, the kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It gives generalizations of nevanlinna theory for meromorphic functions on complete k. The main goal of the notes is to study complex varieties mostly compact or projective.
The rank one symmetric spaces of noncompact type are the real, complex, quaternionic and octonionic hyperbolic spaces. A complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. Free geometry books download ebooks online textbooks tutorials. Many deep problems remain unsolved, some to do with a mysterious connection with arithmetic. But geometry is concerned about the metric, the way things are measured.
Hyperbolicity problems in higherdimensional complex geometry have been intensively studied in recent years. Chapter 15 hyperbolic geometry math 4520, spring 2015 so far we have talked mostly about the incidence structure of points, lines and circles. It is not hard to see that the complex is finitedimensional, but locally infinite. Diy hyperbolic geometry kathryn mann written for mathcamp 2015 abstract and guide to the reader. The arithmetic and the geometry of kobayashi hyperbolicity. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the kobayashi pseudometric is a metric. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the. It is one type of noneuclidean geometry, that is, a geometry that discards one of euclids axioms. It measures the treelikeness of a graph from a metric viewpoint. We also mentioned in the beginning of the course about euclids fifth postulate.
R 1 and consider a symmetric bilinear form of signature n. From the algebraic viewpoint, the set c of complex numbers has the following properties. International conference in nevanlinna theory and complex. The complex of curves on a surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly.
This book presents recent advances on kobayashi hyperbolicity in complex geometry, especially in connection with projective hypersurfaces. Now we study some properties of hyperbolic geometry which do not hold in euclidean geometry. Complex hyperbolic geometry 5 a consequence is a characterization of kobayashi hyperbolicity in terms of isoperimetric inequalities. Math 8 1999 103149 abstract the complex of curves on a surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. Iin complex geometry, we can measure negativity, or hyperbolicity, of a k ahler manifold m. Hyperbolicity in complex geometry harvard university. Hyperbolicity of complex varieties these are notes. Compacti cation and isometries of hyperbolic space 36 2. More precisely, a graph parameter called hyperbolicity, which is related to hyperbolic geometry, is studied in the context of complex networks.
In a saccheri quadrilateral, the summit is longer than the base andthe segment joiningtheir midpoints is shorter than each arm. The arithmetic and the geometry of kobayashi hyperbolicity 3 figure 1. Models there are many other models of ndimensional hyperbolic space. Here the notion of algebraic kobayashi hyperbolicity is as follows. Elliptic, parabolic, and hyperbolic isometries 38 2. In other words, kobayashi hyperbolicity implies brody hyperbolicity. This is a very active field, not least because of the fascinating relations with complex algebraic and arithmetic geometry. Connection between complex hyperbolic and complex trigonometric functions. This leads us to the important concept of a hyperbolic manifold or. Conference on complex geometry, dynamical systems and foliation theory 15 19 may 2017 venue. Hyperbolicity of projective hypersurfaces simone diverio. Hyperbolic geometry is also used to study surface groups.
Equilateral triangle, perpendicular bisector, angle bisector, angle made by lines, the regular hexagon, addition and subtraction of lengths, addition and subtraction of angles, perpendicular lines, parallel lines and angles, constructing parallel. This is a great mathematics book cover the following topics. As complex hyperbolic 1space is just the unit disc in c with the poincar. We give another approach to nevanlinna theory using di. The geometry of knot complements city university of new. Complex geometry, dynamical systems and foliation theory ims. Hyperbolicity problems have a long history and trace back to the small picard theorem and the hyperbolicity of compact riemann surfaces of genus. Complex hyperbolicity is a notion in complex geometry which could be understood either from the point of view of value distribution of entire holomorphic curves in a complex manifold, or the point of view of existence of nonpositive curved metric. Now enters geometry in 1980s, william thurstons seminal work established a strong connection between hyperbolic geometry and knot theory, namely that most knot complements are hyperbolic. However, while there are a number of books on analysis in such spaces, this book is the first to. Let points e and f be the midpoints of the base and summit, respectively. The boundary in complex hyperbolic spaces, known as spherical cr.
This is a set of notes from a 5day doityourself or perhaps discoverityourself introduction to hyperbolic geometry. See figure 5 in 1 for a schematic of how the various projections are related. Intersection numbers and the hyperbolicity of the curve. Weilpetersson metric and hyperbolicity problems of some. Submitted on 21 apr 1998 v1, last revised 11 aug 1998 this version, v2 abstract.
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