Introduction to complex manifolds (Q6539324)

This is a graduate-level textbook on the fundamentals of complex geometry. The prerequisites are knowledge of topological manifolds, smooth manifolds and Riemannian manifolds at the level of J.M. Lee's previous three books published in Graduate Texts in Mathematics (GTM).\N\NComplex manifolds are manifolds with holomorphic coordinates. The depth of the subject comes from the combination of techniques from analysis and differential geometry with the algebraic concepts of sheaves and cohomology. The book presents this theory with detailed attention to both its analytic and algebraic aspects. The topics of bundles and divisors, differential forms and Dolbeault cohomology, sheaves and long exact sequences, and metrics and connections are treated comprehensively.\N\NAs the book develops the theory of complex manifolds, it includes offshoots where general results are applied to derive profound theorems about Riemann surfaces (complex manifolds of complex dimension \(n=1\)). For example, a full proof of the Riemann-Roch theorem is provided, as well as the correspondence between genus \(g=1\) Riemann surfaces and quotients of \(\mathbb{C}\) by a lattice.\N\NThe book culminates with a chapter on Hodge theory and a chapter on the Kodaira embedding theorem. As an application of Hodge theory, the book proves Chow's theorem which states that a complex codimension 1 submanifold of projective space is in fact cut out by the zero locus of a polynomial. The Kodaira embedding theorem is a bridge from the realm of differential geometry and positive Chern curvature tensor to the realm of algebraic geometry and the zero set of homogeneous polynomials.\N\NFull details are always provided. The book includes a very good selection of exercises.\N\NContents:\N\NChapter 1: The Basics\N\NChapter 2: Complex Submanifolds\N\NChapter 3: Holomorphic Vector Bundles\N\NChapter 4: The Dolbeault Complex\N\NChapter 5: Sheaves\N\NChapter 6: Sheaf Cohomology\N\NChapter 7: Connections\N\NChapter 8: Hermitian and Kähler Metrics\N\NChapter 9: Hodge Theory\N\NChapter 10: The Kodaira Embedding Theorem