Algebraic Geometry Over the Complex Numbers

ISBN-13: 9781461418085 / Angielski / Miękka / 2012 / 344 str.

Donu Arapura

Opis

This book provides a rapid introduction to complex algebraic geometry. It combines the study of algebraic geometry with differential and complex geometry and unifies these subjects using sheaf-theoretic ideas.

Recenzje

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 such an 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)

Spis treści

Preface.- 1. Plane Curves.- 2. Manifolds and Varieties via Sheaves.- 3. More Sheaf Theory.- 4. Sheaf Cohomology.- 5. de Rham Cohomoloy of Manifolds.- 6. Riemann Surfaces.- 7. Simplicial Methods.- 8. The Hodge Theorem for Riemann Manifolds.- 9. Toward Hodge Theory for Complex Manifolds.- 10. Kahler Manifolds.- 11. A Little Algebraic Surface Theory.- 12. Hodge Structures and Homological Methods.- 13. Topology of Families.- 14. The Hard Lefschez Theorem.- 15. Coherent Sheaves.- 16. Computation of Coherent Sheaves.- 17. Computation of some Hodge numbers.- 18. Deformation Invariance of Hodge Numbers.- 19. Analogies and Conjectures.- References.- Index.

O autorze

Donu Arapura is a Professor of Mathematics at Purdue University. He received his Ph.D. from Columbia University in 1985. Dr. Arapura’s primary research includes algebraic geometry, and he has written and co-written several publications ranging from Hodge cycles to cohomology.

Blurb

This textbook is a strong addition to existing introductory literature on algebraic geometry. The author’s treatment combines the study of algebraic geometry with differential and complex geometry and unifies these subjects using sheaf-theoretic ideas. It is also an ideal text for showing students the connections between algebraic geometry, complex geometry, and topology, and brings the reader close to the forefront of research in Hodge theory and related fields.

Unique features of this textbook:

- Contains a rapid introduction to complex algebraic geometry

- Includes background material on topology, manifold theory and sheaf theory

- Analytic and algebraic approaches are developed somewhat in parallel The presentation is easy going, elementary, and well illustrated with examples.

“Algebraic Geometry over the Complex Numbers” is intended for graduate level courses in algebraic geometry and related fields. It can be used as a main text for a second semester graduate course in algebraic geometry with emphasis on sheaf theoretical methods or a more advanced graduate course on algebraic geometry and Hodge Theory.