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## About Kodaira’s book on deformations

I happened to read the following sentence in the blog by the physicist Jacques Distler:

"What makes Kodaira’s Complex Manifolds and Deformation of Complex Structures such a delight to read is that he doesn’t neaten up the presentation by removing all the “extraneous” intuitions (both the ones that proved correct, and the ones that didn’t)".

Question: which "extraneous intuitions" (if any) did not prove correct in the book by Kodaira? What could the author of the above phrase be referring to, precisely?

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I seem to remember the book being full of statements like "At this point we expected X to be true and were surprised when we found Y" (in particular in relation to the dimension of the moduli space?). However the book is in my office and I am not. – Jonny Evans Sep 9 2011 at 21:47

You can find lots of these "extraneous" tidbits starting at Chapter 4: Infinitesimal Deformation, where Kodaira pursues what he calls the "main theme" of the book. In particular, the creation of Kodaira--Spencer theory is told kind of like a story. Already on the second page Kodaira mentions how he and Spencer were "rather sceptic" about the "fundamental idea" of the theory, and a few pages later we're told how Kodaira found a particular thing to be "too good to be true" while Spencer held "a more optimistic view" about that same thing.

For a specific example of an intuition that didn't prove correct, let me quote part of the last page of Chapter 4 (in my reprint of the 1986 edition):

Our Theorems 4.2, 4.3, and 4.6 contain the assumption that $\dim H^1(M_t, \Theta_t)$ is independent of $t$. At first we did not know whether this assumption was essential or not. Since we might expect the local triviality of [...], we suspected that we could get rid of this assumption. But the study of deformations of Hopf surfaces revealed the necessity of this assumption.

(Theorems 4.2, 4.3 and 4.6 deal with proving the local triviality of a differentiable family $M_t$ of compact complex manifolds under some certain assumptions.)

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