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Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.

Riemann surfaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Surfaces, Riemann --- Functions --- Congresses --- Differential geometry. Global analysis --- RIEMANN SURFACES --- congresses --- Congresses. --- MATHEMATICS / Calculus. --- Affine space. --- Algebraic function field. --- Algebraic structure. --- Analytic continuation. --- Analytic function. --- Analytic set. --- Automorphic form. --- Automorphic function. --- Automorphism. --- Beltrami equation. --- Bernhard Riemann. --- Boundary (topology). --- Canonical basis. --- Cartesian product. --- Clifford's theorem. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Complex multiplication. --- Conformal geometry. --- Conformal map. --- Coset. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential geometry of surfaces. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euclidean space. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Finsler manifold. --- Fourier series. --- Fuchsian group. --- Function (mathematics). --- Generating set of a group. --- Group (mathematics). --- Hilbert space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Hyperbolic geometry. --- Hyperbolic group. --- Identity matrix. --- Infimum and supremum. --- Inner automorphism. --- Intersection (set theory). --- Intersection number (graph theory). --- Isometry. --- Isomorphism class. --- Isomorphism theorem. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Lorentz group. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix (mathematics). --- Matrix multiplication. --- Measure (mathematics). --- Meromorphic function. --- Metric space. --- Modular group. --- Möbius transformation. --- Number theory. --- Osgood curve. --- Parity (mathematics). --- Partial isometry. --- Poisson summation formula. --- Pole (complex analysis). --- Projective space. --- Quadratic differential. --- Quadratic form. --- Quasiconformal mapping. --- Quotient space (linear algebra). --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemann zeta function. --- Scalar multiplication. --- Scientific notation. --- Selberg trace formula. --- Series expansion. --- Sign (mathematics). --- Square-integrable function. --- Subgroup. --- Teichmüller space. --- Theorem. --- Topological manifold. --- Topological space. --- Uniformization. --- Unit disk. --- Variable (mathematics). --- Riemann, Surfaces de --- RIEMANN SURFACES - congresses --- Fonctions d'une variable complexe --- Surfaces de riemann

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Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Algebraic geometry --- Differential geometry. Global analysis --- 512.7 --- Algebraic geometry. Commutative rings and algebras --- Toric varieties. --- 512.7 Algebraic geometry. Commutative rings and algebras --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Addition. --- Affine plane. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Alexander duality. --- Algebraic curve. --- Algebraic group. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Big O notation. --- Characteristic class. --- Chern class. --- Chow group. --- Codimension. --- Cohomology. --- Combinatorics. --- Commutative property. --- Complete intersection. --- Convex polytope. --- Convex set. --- Coprime integers. --- Cotangent space. --- Dedekind sum. --- Dimension (vector space). --- Dimension. --- Direct proof. --- Discrete valuation ring. --- Discrete valuation. --- Disjoint union. --- Divisor (algebraic geometry). --- Divisor. --- Dual basis. --- Dual space. --- Equation. --- Equivalence class. --- Equivariant K-theory. --- Euler characteristic. --- Exact sequence. --- Explicit formula. --- Facet (geometry). --- Fundamental group. --- Graded ring. --- Grassmannian. --- H-vector. --- Hirzebruch surface. --- Hodge theory. --- Homogeneous coordinates. --- Homomorphism. --- Hypersurface. --- Intersection theory. --- Invertible matrix. --- Invertible sheaf. --- Isoperimetric inequality. --- Lattice (group). --- Leray spectral sequence. --- Limit point. --- Line bundle. --- Line segment. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mixed volume. --- Moduli space. --- Moment map. --- Monotonic function. --- Natural number. --- Newton polygon. --- Open set. --- Picard group. --- Pick's theorem. --- Polytope. --- Projective space. --- Quadric. --- Quotient space (topology). --- Regular sequence. --- Relative interior. --- Resolution of singularities. --- Restriction (mathematics). --- Resultant. --- Riemann–Roch theorem. --- Serre duality. --- Sign (mathematics). --- Simplex. --- Simplicial complex. --- Simultaneous equations. --- Spectral sequence. --- Subgroup. --- Subset. --- Summation. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Toric variety. --- Unit disk. --- Vector space. --- Weil conjecture. --- Zariski topology.

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The classical uniformization theorem for Riemann surfaces and its recent extensions can be viewed as introducing special pseudogroup structures, affine or projective structures, on Riemann surfaces. In fact, the additional structures involved can be considered as local forms of the uniformizations of Riemann surfaces. In this study, Robert Gunning discusses the corresponding pseudogroup structures on higher-dimensional complex manifolds, modeled on the theory as developed for Riemann surfaces.Originally published in 1978.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Analytical spaces --- Differential geometry. Global analysis --- Complex manifolds --- Connections (Mathematics) --- Pseudogroups --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Global analysis (Mathematics) --- Lie groups --- Geometry, Differential --- Analytic spaces --- Manifolds (Mathematics) --- Adjunction formula. --- Affine connection. --- Affine transformation. --- Algebraic surface. --- Algebraic torus. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Automorphic function. --- Automorphism. --- Bilinear form. --- Canonical bundle. --- Characterization (mathematics). --- Cohomology. --- Compact Riemann surface. --- Complex Lie group. --- Complex analysis. --- Complex dimension. --- Complex manifold. --- Complex multiplication. --- Complex number. --- Complex plane. --- Complex torus. --- Complex vector bundle. --- Contraction mapping. --- Covariant derivative. --- Differentiable function. --- Differentiable manifold. --- Differential equation. --- Differential form. --- Differential geometry. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Elliptic operator. --- Elliptic surface. --- Enriques surface. --- Equation. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exterior derivative. --- Fiber bundle. --- General linear group. --- Geometric genus. --- Group homomorphism. --- Hausdorff space. --- Holomorphic function. --- Homomorphism. --- Identity matrix. --- Invariant subspace. --- Invertible matrix. --- Irreducible representation. --- Jacobian matrix and determinant. --- K3 surface. --- Kähler manifold. --- Lie algebra representation. --- Lie algebra. --- Line bundle. --- Linear equation. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Manifold. --- Mathematical analysis. --- Mathematical induction. --- Ordinary differential equation. --- Partial differential equation. --- Permutation. --- Polynomial. --- Principal bundle. --- Projection (linear algebra). --- Projective connection. --- Projective line. --- Pseudogroup. --- Quadratic transformation. --- Quotient space (topology). --- Representation theory. --- Riemann surface. --- Riemann–Roch theorem. --- Schwarzian derivative. --- Sheaf (mathematics). --- Special case. --- Subalgebra. --- Subgroup. --- Submanifold. --- Symmetric tensor. --- Symmetrization. --- Tangent bundle. --- Tangent space. --- Tensor field. --- Tensor product. --- Tensor. --- Theorem. --- Topological manifold. --- Uniformization theorem. --- Uniformization. --- Unit (ring theory). --- Vector bundle. --- Vector space. --- Fonctions de plusieurs variables complexes --- Variétés complexes

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This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids "ations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.

CR submanifolds. --- Dehn surgery (Topology). --- Three-manifolds (Topology). --- CR submanifolds --- Dehn surgery (Topology) --- Three-manifolds (Topology) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Cauchy-Riemann submanifolds --- Submanifolds, CR --- Low-dimensional topology --- Topological manifolds --- Surgery (Topology) --- Manifolds (Mathematics) --- Arc (geometry). --- Automorphism. --- Ball (mathematics). --- Bijection. --- Bump function. --- CR manifold. --- Calculation. --- Canonical basis. --- Cartesian product. --- Clifford torus. --- Combinatorics. --- Compact space. --- Conjugacy class. --- Connected space. --- Contact geometry. --- Convex cone. --- Convex hull. --- Coprime integers. --- Coset. --- Covering space. --- Dehn surgery. --- Dense set. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differential geometry of surfaces. --- Discrete group. --- Double coset. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Equivalence relation. --- Euclidean distance. --- Four-dimensional space. --- Function (mathematics). --- Fundamental domain. --- Geometry and topology. --- Geometry. --- Harmonic function. --- Hexagonal tiling. --- Holonomy. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Horosphere. --- Hyperbolic 3-manifold. --- Hyperbolic Dehn surgery. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperbolic triangle. --- Hypersurface. --- I0. --- Ideal triangle. --- Intermediate value theorem. --- Intersection (set theory). --- Isometry group. --- Isometry. --- Limit point. --- Limit set. --- Manifold. --- Mathematical induction. --- Metric space. --- Möbius transformation. --- Parameter. --- Parity (mathematics). --- Partial derivative. --- Partition of unity. --- Permutation. --- Polyhedron. --- Projection (linear algebra). --- Projectivization. --- Quotient space (topology). --- R-factor (crystallography). --- Real projective space. --- Right angle. --- Sard's theorem. --- Seifert fiber space. --- Set (mathematics). --- Siegel domain. --- Simply connected space. --- Solid torus. --- Special case. --- Sphere. --- Stereographic projection. --- Subgroup. --- Subsequence. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Topology. --- Torus. --- Transversality (mathematics). --- Triangle group. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Unit tangent bundle.

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In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure. The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinement of the toroidal compactifications by Mumford et al. For general D, fine moduli spaces may have slits caused by Griffiths transversality at the boundary and be no longer locally compact. Second, Kato and Usui construct eight enlargements of D and describe their relations by a fundamental diagram, where four of these enlargements live in the Hodge theoretic area and the other four live in the algebra-group theoretic area. These two areas are connected by a continuous map given by the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the construction in the first topic.

Hodge theory. --- Logarithms. --- Logs (Logarithms) --- Algebra --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Algebraic group. --- Algebraic variety. --- Analytic manifold. --- Analytic space. --- Annulus (mathematics). --- Arithmetic group. --- Atlas (topology). --- Canonical map. --- Classifying space. --- Coefficient. --- Cohomology. --- Compactification (mathematics). --- Complex manifold. --- Complex number. --- Congruence subgroup. --- Conjecture. --- Connected component (graph theory). --- Continuous function. --- Convex cone. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential form. --- Direct image functor. --- Divisor. --- Elliptic curve. --- Equivalence class. --- Existential quantification. --- Finite set. --- Functor. --- Geometry. --- Hodge structure. --- Homeomorphism. --- Homomorphism. --- Inverse function. --- Iwasawa decomposition. --- Local homeomorphism. --- Local ring. --- Local system. --- Logarithmic. --- Maximal compact subgroup. --- Modular curve. --- Modular form. --- Moduli space. --- Monodromy. --- Monoid. --- Morphism. --- Natural number. --- Nilpotent orbit. --- Nilpotent. --- Open problem. --- Open set. --- P-adic Hodge theory. --- P-adic number. --- Point at infinity. --- Proper morphism. --- Pullback (category theory). --- Quotient space (topology). --- Rational number. --- Relative interior. --- Ring (mathematics). --- Ring homomorphism. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Smooth morphism. --- Special case. --- Strong topology. --- Subgroup. --- Subobject. --- Subset. --- Surjective function. --- Tangent bundle. --- Taylor series. --- Theorem. --- Topological space. --- Topology. --- Transversality (mathematics). --- Two-dimensional space. --- Vector bundle. --- Vector space. --- Weak topology.