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Differential analysis on complex manifolds. With a new appendix by Oscar Garcia-Prada. 3rd ed. (English) Zbl 1131.32001

Graduate Texts in Mathematics 65. New York, NY: Springer (ISBN 978-0-387-73891-8/hbk). xiii, 299 p. (2008).
The first edition of this book appeared in 1973 [Differential analysis on complex manifolds. (Englewood Cliffs), N.J.: Prentice-Hall, Inc. X, (1973; Zbl 0262.32005)]. The book was an outgrowth with some expansions of the lectures given by the author at Brandeis University in 1967–1968 and Rice University in 1968–1969.
The second edition of the book appeared in 1980 with the same title in the collection [Graduate Texts in Mathematics. 65. (New York-Heidelberg-Berlin): Springer-Verlag (1980; Zbl 0435.32004)]. This second edition contained a new section on the classical finite-dimensional representation theory for \({\mathfrak{sl}}(2,{\mathbb C})\) and an application in the proof of the Lefschetz decomposition theorem. The content of the book was generally the same as the first edition with several corrections to some errors present in the first edition and with some new examples added throughout.
The third edition contains the same six chapters from the first edition and an appendix “Moduli spaces and geometric structures” written by Oscar Garcia-Prada. The purpose of the text is to present the basics of analysis and geometry on compact complex manifolds and is already one of the standard sources for this material.
In the first two chapters there are presented some topics on complex manifolds and vector bundles, presheaves and sheaves as well as some elements of cohomology with coefficients in a sheaf. In the third chapter the author presents some notions and results from the Hermitian differential geometry, the canonical connection of a Hermitian vector bundle, Chern classes and complex line bundles. In the fourth chapter with the title “Elliptic operator theory”, there are presented the Sobolev spaces, differential and pseudodifferential operators, and elliptic complexes. After studying some special topics about compact complex manifolds (harmonic theory on compact manifolds, representations of \({\mathfrak{sl}}(2,{\mathbb{C}})\) on Hermitian exterior algebras, differential operators on Kähler manifolds, Hodge-Riemann bilinear relations on Kähler manifolds) the author presents in the last chapter, Kodaira’s projective embedding theorem.
In the appendix (having about 40 pages), Oscar Garcia-Prada presents some topics related to vector and Higgs bundles on Riemann surfaces, representations of the fundamental group, non-Abelian Hodge theory, representations in \(\mathbb{U}(p,q)\) and Higgs bundles, moment maps and moduli spaces, higher dimensional generalizations.
The book has proven to be an excellent introduction to the theory of complex manifolds considered from both the points of view of complex analysis and differential geometry.

MSC:

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
32Q99 Complex manifolds
58A14 Hodge theory in global analysis
14E25 Embeddings in algebraic geometry
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