At a Glance
344 Pages
23.39 x 15.6 x 1.83
Paperback
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Industry Reviews
From the reviews:
"The book under review evolved from various courses in algebraic geometry the author taught at Purdue University. It is intended for graduate level courses on algebraic geometry over C. ... Every section of each chapter ends with a series of exercises that complement the treated material, sometimes asking to give proofs of stated results. ... This work can serve as a textbook in an introductory course in algebraic geometry with a strong emphasis on its transcendental aspects, or as a reference book on the subject." (Pietro De Poi, Mathematical Reviews, June, 2013)
"Masterful mathematical expositors guide readers along a meaningful journey. ... Every student should read this book first before grappling with any of those bibles. ... This is an advanced book in its own right ... . Arapura's knack for doing things in the simplest possible way and explaining the 'why' makes for much easier reading than one might reasonably expect. Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 50 (5), January, 2013)
"The book under review is a welcome addition to the literature on complex algebraic geometry. The approach chosen by the author balances the algebraic and transcendental approaches and unifies them by using sheaf theoretical methods. ... This is a well-written text ... with plenty of examples to illustrate the ideas being discussed." (Felipe Zaldivar, The Mathematical Association of America, June, 2012)
"Book provides a very lucid, vivid, and versatile first introduction to algebraic geometry, with strong emphasis on its transcendental aspects. The author provides a broad panoramic view of the subject, illustrated with numerous instructive examples and interlarded with a wealth of hints for further reading. Indeed, the balance between rigor, intuition, and completeness in the presentation of the material is absolutely reasonable for suchan introductory course book, and ... it may serve as an excellent guide to the great standard texts in the field." (Werner Kleinert, Zentralblatt MATH, Vol. 1235, 2012)
Preface | p. vii |
Introduction through Examples | |
Plane Curves | p. 3 |
Conics | p. 3 |
Singularities | p. 5 |
Bézout's Theorem | p. 7 |
Cubics | p. 9 |
Genus 2 and 3 | p. 11 |
Hyperelliptic Curves | p. 14 |
Sheaves and Geometry | |
Manifolds and Varieties via Sheaves | p. 21 |
Sheaves of Functions | p. 22 |
Manifolds | p. 24 |
Affine Varieties | p. 28 |
Algebraic Varieties | p. 32 |
Stalks and Tangent Spaces | p. 35 |
1-Forms, Vector Fields, and Bundles | p. 41 |
Compact Complex Manifolds and Varieties | p. 45 |
More Sheaf Theory | p. 49 |
The Category of Sheaves | p. 49 |
Exact Sequences | p. 53 |
Affine Schemes | p. 58 |
Schemes and Gluing | p. 62 |
Sheaves of Modules | p. 66 |
Line Bundles on Projective Space | p. 70 |
Direct and Inverse Images | p. 72 |
Differentials | p. 76 |
Sheaf Cohomology | p. 79 |
Flasque Sheaves | p. 79 |
Cohomology | p. 81 |
Soft Sheaves | p. 86 |
C∞-Modules Are Soft | p. 89 |
Mayer-Vietoris Sequence | p. 90 |
Products* | p. 93 |
De Rham Cohomology of Manifolds | p. 97 |
Acyclic Resolutions | p. 97 |
De Rham's Theorem | p. 100 |
Künneth's Formula | p. 102 |
Poincaré Duality | p. 105 |
Gysin Maps | p. 108 |
Projections | p. 109 |
Inclusions | p. 110 |
Fundamental Class | p. 111 |
Lefschetz Trace Formula | p. 113 |
Riemann Surfaces | p. 117 |
Genus | p. 117 |
∂-Cohomology | p. 122 |
Projective Embeddings | p. 126 |
Function Fields and Automorphisms | p. 130 |
Modular Forms and Curves | p. 133 |
Simphicial Methods | p. 137 |
Simplicial and Singular Cohomology | p. 137 |
Cohomology of Projective Space | p. 142 |
Cech Cohomology | p. 144 |
Cech Versus Sheaf Cohomology | p. 147 |
First Chern Class | p. 150 |
Hodge Theory | |
The Hodge Theorem for Riemannian Manifolds | p. 157 |
Hodge Theory on a Simplicial Complex | p. 157 |
Harmonic Forms | p. 159 |
The Heat Equation* | p. 163 |
Toward Hodge Theory for Complex Manifolds | p. 169 |
Riemann Surfaces Revisited | p. 169 |
Dolbeault's Theorem | p. 172 |
Complex Tori | p. 173 |
Kähler Manifolds | p. 179 |
Kähler Metrics | p. 179 |
The Hodge Decomposition | p. 183 |
Picard Groups | p. 187 |
A Little Algebraic Surface Theory | p. 189 |
Examples | p. 189 |
The Neron-Severi Group | p. 193 |
Adjunction and Riemann-Roch | p. 195 |
The Hodge Index Theorem | p. 198 |
Fibered Surfaces* | p. 200 |
Hodge Structures and Homological Methods | p. 203 |
Pure Hodge Structures | p. 203 |
Canonical Hodge Decomposition | p. 205 |
Hodge Decomposition for Moishezon Manifolds | p. 210 |
Hypercohomology* | p. 212 |
Holomorphic de Rham Complex* | p. 216 |
The Deligne-Hodge Decomposition* | p. 217 |
Topology of Families | p. 223 |
Topology of Families of Elliptic Curves | p. 223 |
Local Systems | p. 228 |
Higher Direct Images* | p. 230 |
First Betti Number of a Fibered Variety* | p. 235 |
The Hard Lefschetz Theorem | p. 237 |
Hard Lefschetz | p. 237 |
Proof of Hard Lefschetz | p. 239 |
Weak Lefschetz and Barth's Theorem | p. 241 |
Lefschetz Pencils* | p. 242 |
Cohomology of Smooth Projective Maps* | p. 247 |
Coherent Cohomology | |
Coherent Sheaves | p. 255 |
Coherence on Ringed Spaces | p. 255 |
Coherent Sheaves on Affine Schemes | p. 257 |
Coherent Sheaves on Pn | p. 259 |
GAGA, Part I | p. 263 |
Cohomology of Coherent Sheaves | p. 265 |
Cohomology of Affine Schemes | p. 265 |
Cohomology of Coherent Sheaves on Pn | p. 267 |
Cohomology of Analytic Sheaves | p. 272 |
GAGA, Part II | p. 274 |
Computation of Some Hodge Numbers | p. 279 |
Hodge Numbers of Pn | p. 279 |
Hodge Numbers of a Hypersurface | p. 282 |
Hodge Numbers of a Hypersurface II | p. 285 |
Double Covers | p. 287 |
Griffiths Residues* | p. 289 |
Deformations and Hodge Theory | p. 293 |
Families of Varieties via Schemes | p. 293 |
Semicontinuity of Coherent Cohomology | p. 297 |
Deformation Invariance of Hodge Numbers | p. 300 |
Noether-Lefschetz* | p. 302 |
Analogies and Conjectures* | |
Analogies and Conjectures | p. 307 |
Counting Points and Euler Characteristics | p. 307 |
The Weil Conjectures | p. 309 |
A Transcendental Analogue of Weil's Conjecture | p. 312 |
Conjectures of Grothendieck and Hodge | p. 313 |
Problem of Computability | p. 317 |
Hodge Theory without Analysis | p. 319 |
References | p. 321 |
Index | p. 327 |
Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9781461418085
ISBN-10: 1461418089
Series: Universitext
Published: 1st February 2012
Format: Paperback
Language: English
Number of Pages: 344
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: US
Dimensions (cm): 23.39 x 15.6 x 1.83
Weight (kg): 0.48
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