Are there any good introductory type of books that is focus on complex manifolds? Thanks.
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1$\begingroup$ I would suggest Hirzebruch's "Topological methods in algebraic geometry". $\endgroup$– Alex DegtyarevCommented Nov 9, 2014 at 11:41
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2$\begingroup$ Matsushima's"Differentiable Manifolds" if you jus want the basics. For deformation of complex manifolds I like "Advances in Moduli theory" by Shimizu. But if you want all modern techniques you will need Griffith and Harris'"Principles of Algebraic Geometry" or www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf $\endgroup$– user40276Commented Nov 9, 2014 at 11:53
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4 Answers
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Huybrechts Complex Geometry is excellent, and has some more recent stuff.
Griffiths and Harris Principles of Algebraic Geometry is a great classic.
Barths, Peters and Van Den Ven Compact Complex Surfaces gives a nice explanation of the classification of surfaces, which gives lots of nice examples, including nonalgebraic ones.
Beauville, Complex Algebraic Surfaces covers the classification of surfaces in the algebraic category.
Demailly, Complex Analytic and Differential Geometry is more comprehensive, from the transcendental point of view.
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MR2359489 Wells, Raymond O., Jr. Differential analysis on complex manifolds. With a new appendix by Oscar Garcia-Prada. Graduate Texts in Mathematics, 65. Springer, New York, 2008.
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I would suggest "Complex Manifolds" by James Morrow and Kunihiko Kodaira. Kodaria is a very good lecturer who explains things really well.
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I recommend having two or three books around. Sometimes you understand one book's explanation of a topic better than the others. I also found it helpful to read a book on Riemann surfaces, such as Gunning or Springer at the same time.