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Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics) [Paperback]

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Item description for Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics) by Kunihiko Kodaira...

From the reviews:

"The author, [...], has written a book which will be of service to all who are interested in this by now vast subject. [...] 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

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Item Specifications...

Pages   465
Est. Packaging Dimensions:   Length: 9.1" Width: 6.2" Height: 0.9"
Weight:   1.6 lbs.
Binding  Softcover
Release Date   Dec 22, 2004
Publisher   Springer
ISBN  3540226141  
ISBN13  9783540226147  

Availability  139 units.
Availability accurate as of Oct 28, 2016 10:14.
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More About Kunihiko Kodaira

Register your artisan biography and upload your photo! Kunihiko Kodaira (1915-1997) worked in many areas including harmonic integrals, algebraic geometry and the classification of compact complex analytic surfaces. He held faculty positions at many universities including Tokyo, Harvard, Stanford, and Johns Hopkins, and the Institute for Advanced Study in Princeton. He was awarded a Fields medal in 1954 and a Wolf Prize in 1984.

Kunihiko Kodaira was born in 1915.

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Reviews - What do customers think about Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics)?

A Japanese mathematician  Mar 1, 2006
Kunihiko Kodaira is one of the greatest japanese mathematicians.

He is one of the last students of Teiji Takagi who is famous
for class field theory.

Kunihiko Kodaira died at Kohu, Yamanashi prefecture, Japan in 1997.

He also wrote essays.
Superb  Sep 12, 2005
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.'

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