By Kunihiko Kodaira

*From the reviews:*

"The writer, who with Spencer created the idea of deformations of a fancy manifold, has written a booklet for you to be of provider to all who're attracted to this by means of now massive topic. even if meant for a reader with a undeniable mathematical adulthood, the writer starts off in the beginning, [...]. it is a e-book of many virtues: mathematical, historic, and pedagogical. components of it may be used for a graduate complicated manifolds course."

J.A. Carlson in *Mathematical Reviews*, 1987

"There are many mathematicians, or maybe physicists, who might locate this publication precious and available, yet its distinct characteristic is the perception it supplies right into a terrific mathematician's paintings. [...] it's interesting to experience among the traces Spencer's optimism, Kodaira's scepticism or the shadow of Grauert along with his very diverse equipment, because it is to listen to of the surprises and ironies which seemed at the means. so much of all it's a piece of labor which indicates arithmetic as mendacity someplace among discovery and invention, a truth which all mathematicians understand, yet so much inexplicably cover of their work."

N.J. Hitchin within the *Bulletin of the London Mathematical Society*, 1987

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**Additional info for Complex Manifolds and Deformation of Complex Structures**

**Example text**

C„) ^ ( 0 , . . , 0) of natural numbers, there are no non-constant meromorphic functions on M = WIG. (c) Surgery We explain another method to construct complex manifolds. Let M be a complex manifold, and 5 c M a compact submanifold of M. We construct a new complex manifold M = {M — S)u S by replacing S by another compact complex manifold S as follows: Take domains W and W^ such that 5 c: Wi c [ \ ^ J c: \y c: M, where [ W] is assumed to be compact. Let 5 be a compact complex submanifold of a complex manifold W such that there are a domain W] with 5 c: W, c [ W'j] c: W, and a bihojomorphic map O of W - 5 onto W - 5 such that ^{W^-S) = W^-S.

By local coordinates with centre q we mean a biholomorphic map Zq\ p-^ Zq{p) of a domain U{q) containing q onto a domain of C" containing ( 0 , . . , 0) such that Zq{q) = 0. Let Zq'. , z''q{p)) bc givcn local coordinates with centre q. 1. (0) = {z|z = ( z ^ . . , z J , | z ^ | < r ^ . . , | z " | < r " } , such that [(7,(0)] c:z^((7(^)), where r = {r\ ... r"") and r ' > O f o r all /. 1. Let M"" he a complex manifold. Suppose that for each point qe M, local coordinates z^ with centre q and a coordinate polydisk UjK^q^iq) are given.

I Corollary. Let f{z) he a holomorphic function in a domain of C", and S the analytic hypersurface defined by the equation f{z) = 0. We assume OeS. Supppose fo{z) = 0 is a minimal equation of S at 0. Then if e>0 is small enough, fiz) = 0 is a minimal equation ofS at cfor every ce S with \c\ < s. 14. Let U he a. 17, if fiz) = 0 and g{z) = 0 are both minimal equations of S in U, u{z) = f(z)/g{z) is a non-vanishing holomorphic function in U. 2. Holomorphic Map In this section we consider a map O: z^w = <^(z) of a domain D c C " into C".