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Differential Analysis on Complex Manifolds

Raymond O. Wells, Jr.
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 65
[Reviewed by
Fernando Q. Gouvêa
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This is the third edition of a well-known book first published in 1973, with a second edition in 1980. When the second edition appeared, Mathematical Reviews said that it was "already one of the standard sources for this material." It is good to see it back 28 years later.

Wells' book is an example of the "straight path to big theorem" approach to mathematical exposition. He has chosen two big results, namely the Hodge and Lefschetz decompositions for the cohomology of a compact Kähler manifold and Kodaira's vanishing and projective embedding theorems for Hodge manifolds. The book is written with a view to getting to these results as efficiently as possible. This has both advantages and disadvantages. For someone learning the material for the first time (or for a professor planning a series of lectures), having such a goal in mind often serves as motivation and gives coherence to the material. On the other hand, readers hoping for broad coverage of the material will need to look elsewhere.

Rather than rewrite the book to take into account what has happened since the original editions, Wells has asked Oscar García-Prada to write an appendix. Entitled "Moduli Spaces and Geometric Structures," it focuses on the moduli theory of vector bundles and Higgs bundles on Riemann surfaces and its connections to mathematical physics. The appendix is 40 pages long and only loosely integrated with the rest of the book; for example, it has its own bibliography. García-Prada has written in an expository style, "in a rather sketchy way, avoiding many technical aspects and omitting most proofs."

Manifolds and Vector Bundles * Sheaf Theory * Differential Geometry * Elliptic Operator Theory * Compact Complex Manifolds * Kodaira’s Projective Embedding Theorem * Appendix by O. Garcia-Prada * References * Subject Index * Author Index