By James Morrow and Kunihiko Kodaira

This quantity serves as an advent to the Kodaira-Spencer thought of deformations of complicated constructions. according to notes taken via James Morrow from lectures given by means of Kunihiko Kodaira at Stanford collage in 1965-1966, the ebook offers the unique evidence of the Kodaira embedding theorem, exhibiting that the limited classification of Kähler manifolds referred to as Hodge manifolds is algebraic. incorporated are the semicontinuity theorems and the neighborhood completeness theorem of Kuranishi. Readers are assumed to understand a few algebraic topology. whole references are given for the implications which are used from elliptic partial differential equations. The publication is acceptable for graduate scholars and researchers drawn to summary advanced manifolds.

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Uf(aj) (') Ug(a j ) A. Let IXo, ••• , IXq E A. (')... 1 (')... /"-... U9(II/) (') ... (') Ug(aq) , and Ui = Uf(II,) (') ... , g: A ~ J are two refining maps. Define a function (kU)A, ... A. by q (kuh, ... A. = p"f;o ( -1)P- 1 rV ouf(A,)· .. ) (5) Let us call the maps n~, defined by f and g,f*, and g*. We claim that the following equation holds: [(ok + (6) kO)]IIo ... a. = (g*u - 1*u)ao ... a•. The function ku is not necessarily skew-symmetric in its indices; so we skew-symmetrize -r~I .. · A = (k'-r)A' ...

J In Example (4) 9' = X x C with the following topology: Let s = (x, z); then OJI(s) = {(y, z) lyE U, z fixed}. If r -+ f(r) is a continuous map into Y of 1= {r Ia < r < b}, then f(l) = {(y, z) I z fixed and y = w(f(r»r E l}. In other words we give X x C the product topology where X has its given topology and C has the discrete topology. 2. Let U be a subset (usually open) of X. By a section u of 9' over U we mean a continuous map x --. u(x) such that iii u(x) = x. J (or ~). If f(z) is a holomorphic (or differentiable) function on U, then u: p -+ f p , p E U is a section.

2. h([/) is a subsheaf of [/". The proofs are left to the reader.