By Kunihiko Kodaira

*From the reviews:*

"The writer, who with Spencer created the speculation of deformations of a fancy manifold, has written a publication in an effort to be of provider to all who're drawn to this through now tremendous topic. even supposing meant for a reader with a undeniable mathematical adulthood, the writer starts in the beginning, [...]. it is a ebook of many virtues: mathematical, historic, and pedagogical. components of it can be used for a graduate advanced manifolds course."

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

"There are many mathematicians, or maybe physicists, who may locate this publication valuable and obtainable, yet its exact characteristic is the perception it supplies right into a marvelous mathematician's paintings. [...] it really is interesting to experience among the traces Spencer's optimism, Kodaira's scepticism or the shadow of Grauert together with his very assorted 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 be aware of, yet so much inexplicably disguise 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**

Z^) T^ 0. 19, 0 : ( z j , . . , z„) is a biholomorphic map of U{z^) onto a neighbourhood (7( w^) of w'=:

Identifying Uj with % as usual, we may consider M = U^ %- Thus a compact complex manifold M is obtained by glueing a finite number of domains ^ i , . . , ^^^ inC" via the identification o f Zfc G %L^j CZ OU. w i t h Zj = T,fc(Zfc) E %k C ^fc. A holomorphic function defined on a compact complex manifold M is a constant, Proof Suppose that/(;7) is holomorphic on all of M. Since M is compact, the continuous function \fip)\ attains its maximum at some point qeM. Let q e Uj, and put f(p) =fj{Zj) on Uj where Zj is a local coordinate system on Uj.

X 6 ) = det = det a ( z i , Z2, Z3, z i , Z2, Z3) a ( W l , W2, W3) (9(Wi, M>2, W3) a ( z i , Z2, Z3) (9(zi, Z2, Z3) = |/(z)P. 2. Holomorphic Map 25 Let ^ : z-» w = <|)(z) be a holomorphic map of D c C " into C", and ^ : w ^ ^ = ^ ( w ) a holomorphic map of E c z C " into C". If 0 ( D ) c : £, the composite '^ ° 0 : z -> f = '^(