Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics) 2005th Edition by Kunihiko Kodaira (PDF)

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Kodaira is a Fields Medal Prize Winner. (In the absence of a Nobel prize in mathematics, they are regarded as the highest professional honour a mathematician can attain.)Kodaira is an honorary member of the London Mathematical Society.Affordable softcover edition of 1986 classic

User’s Reviews

Editorial Reviews: From the Back Cover From the reviews:”The author, who with Spencer created the theory of deformations of a complex manifold, has written a book which will be of service to all who are interested in this by now vast subject. Although intended for a reader with a certain mathematical maturity, the author begins at the beginning, […]. This is a book of many virtues: mathematical, historical, and pedagogical. Parts of it could be used for a graduate complex manifolds course.”J.A. Carlson in Mathematical Reviews, 1987″There are many mathematicians, or even physicists, who would find this book useful and accessible, but its distinctive attribute is the insight it gives into a brilliant mathematician’s work. […] It is intriguing to sense between the lines Spencer’s optimism, Kodaira’s scepticism or the shadow of Grauert with his very different methods, as it is to hear of the surprises and ironies which appeared on the way. Most of all it is a piece of work which shows mathematics as lying somewhere between discovery and invention, a fact which all mathematicians know, but most inexplicably conceal in their work.”N.J. Hitchin in the Bulletin of the London Mathematical Society, 1987 About the Author Kunihiko Kodaira was born on March 16, 1915 in Tokyo, Japan. He graduated twice from the University of Tokyo, with a degree in mathematics in 1938 and one in physics in 1941. From 1944 until 1949, Kodaira was an associate professor at the University of Tokyo but by this time his work was well known to mathematicians worldwide and in 1949 he accepted an invitation from H. Weyl to come to the Institute for Advanced Study. During his 12 years in Princeton, he was also Professor at Princeton University from 1952 to 1961. After a year at Harvard, he was then appointed in 1962 to the chair of mathematics at Johns Hopkins University, which he left in 1965 for a chair at Stanford University. Finally, after 2 years at Stanford, he returned to Japan to Tokyo University from 1967. He died in Kofu, Japan, in 1997.Kodaira’s work covers many topics, including applications of Hilbert space methods to differential equations, harmonic integrals , and importantly the application of sheaves to algebraic geometry. Around 1960 he became involved in the classification of compact complex analytic spaces. One of the themes running through much of his work is the Riemann-Roch theorem and this played an important role in much of his research.Kodaira received many honours for his outstanding research, in particular the Fields Medal, in 1954.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐If you are doing complex differential or algebraic geometry, and you are feeling ungrounded after working out of Wells or Griffiths and Harris, spend some time with this book. Kodaira manages to be concrete about the subject, in a way that I found very helpful to developing intuition. Also, if you are studying Teichmuller theory or moduli of curves, and the deformation theory seems perversely obfuscated in other texts, this is the book to turn to.

⭐Classic!

⭐A very good book, flawlessly delivered.

⭐I think this is an excellent book for beginners who want to study complex geometry.

⭐Of importance to applications such as superstring theories in high-energy physics, the theory of complex manifolds and the deformation of complex structures are explained in great detail in this book by one of the major contributors to the subject. One of the valuable features of the book that is actually rare in more recent books on mathematics is that the author tries (and succeeds) to give motivation for the subject. This feature is actually quite common in older books on mathematics, for with few exceptions writers at that time believed that a proper understanding of mathematics can only come with explanations that are given outside the deductive structures that are created in the process of doing mathematics. These explanations frequently involve the use of diagrams, pictures, intuitive arguments, and historical analogies, and so are not held to be rigorous from a mathematical standpoint. They are however extremely valuable to students of mathematics and those who are interested in applying it, like physicists and engineers. There seems to be an inverse relationship between rigor and understanding of mathematics, and given the emphasis on the former in modern works of mathematics, one can expect students to have more trouble learning a particular branch of mathematics than those students of a few decades ago.Luckily though the author of this book has given the reader valuable insights into the nature of complex manifolds and what is means to deform a complex structure. Complex manifolds are different from real manifolds due to the notion of holomorphicity, but are similar in the sense that they are constructed from domains that are “glued together”. In complex manifolds, the “glue” is provided by biholomorphic maps between the domains, the latter of which are open sets called `polydisks’. A `deformation’ of the complex manifold is then considered to be a glueing of the same polydisks but via a different identification. For an n-dimensional complex manifold, the maps could thus be dependent on say m parameters, which are labeled as “t” by the author. This dependence on t would result in a differentiable family of complex manifolds. One thus expects the complex manifold to be dependent on t, but the author discusses a counterexample that indicates that one must not be cavalier about this approach.The definition that is arrived at involves letting t be an element of a domain B in m-dimensional Euclidean space, and considering a collection of compact complex n-dimensional manifolds that depends on t. This collection will be a `differentiable family’ if: 1. There exists a differentiable manifold M and a C-infinity map W from M onto B such that the rank of the Jacobian matrix of W is equal to m at every point of M. 2. M(t), the inverse image of t under W is a compact connected subset of M, and in fact is equal to a member of the collection. 3. M has a locally finite open covering along with smooth coordinate functions on the covering that have non-empty intersection with each member of the covering. Beginning with an initial element of B, each member of the inverse image of t under W is viewed as a deformation of the initial member. The crucial point made by the author is that the restricting the domain of the parameter t to a sufficiently small interval allows the representation of the member M(t) as a union of polydisks that are independent of t. Therefore only the coordinate transformations depend on t, and thus only the way of glueing the polydisks depends on t.To show that these constructions are meaningful, namely that the complex structure of M(t) actually depends on t, the author studies the case of m = 1. In the process he constructs the infinitesimal deformation of M(t), and interprets it as the derivative of the complex structure of M(t) with respect to t. He also shows that the infinitesimal deformation does not depend on the choice of systems of local coordinates, and that the infinitesimal deformation vanishes when M(t) does not vary with t. The author then defines, using a notion of equivalence between two differentiable families, a differentiable family (M, B, W) to be `trivial’ if it is equivalent to a product (M x B, B, P). Restricting this triviality to a subdomain gives a notion of `local triviality’, which implies immediately that each M(t) will be biholomorphically equivalent to a fixed M. He then shows that if the infinitesimal deformation vanishes then the differentiable family is locally trivial. A more substantial statement of this result is encapsulated in the Frolicher-Nijenhuis theorem, which follows from the results that the author proves in the book. These results involve the theory of strongly elliptic differential operators and considerations of the first cohomology group of M(t) with coefficients in the sheaf of germs of holomorphic vector fields over M(t).The case of a complex analytic family of compact complex manifolds entails that B will be domain in complex n-space. The author shows that a complex analytic family will be trivial if it is trivial as a differentiable family. As expected, because of the nature of analyticity learned from the theory of complex variables, the proof of these results involves the theory of harmonic differential forms. The author gives these proofs in detail in the book. He also considers the question whether if given an element b of the first cohomology group of a compact complex manifold Mwith coefficients in the sheaf of germs of holomorphic vector fields over M, one can find a complex analytic family that takes M as its initial element and the derivative equal to b. This question, as expected, involves the use of obstruction theory, which the author develops in great detail. In these considerations, the reader will see the and origin and role of the moduli of complex structures. These are essentially the number of parameters m, as long as the complex analytic family is `complete.,The deformations theory is equivalent to that of foliations. We can consider little variations of the curves whom we have got. That can be posed in a model constructed with the lifting, which is an important concept of the tradition. So it born the foliations, which are made of leaves. All that can entry also in the context of orbits.

⭐Kunihiko Kodaira is one of the greatest japanese mathematicians.He is one of the last students of Teiji Takagi who is famousfor class field theory.Kunihiko Kodaira died at Kohu, Yamanashi prefecture, Japan in 1997.He also wrote essays.

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